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Theorem wemapwe 8579
Description: Construct lexicographic order on a function space based on a reverse well-ordering of the indexes and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.)
Hypotheses
Ref Expression
wemapwe.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
wemapwe.u 𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}
wemapwe.2 (𝜑𝑅 We 𝐴)
wemapwe.3 (𝜑𝑆 We 𝐵)
wemapwe.4 (𝜑𝐵 ≠ ∅)
wemapwe.5 𝐹 = OrdIso(𝑅, 𝐴)
wemapwe.6 𝐺 = OrdIso(𝑆, 𝐵)
wemapwe.7 𝑍 = (𝐺‘∅)
Assertion
Ref Expression
wemapwe (𝜑𝑇 We 𝑈)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦   𝑤,𝐹,𝑥,𝑦,𝑧   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑤,𝑅,𝑧   𝑧,𝑆   𝑥,𝑈,𝑦   𝑥,𝑍
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐵(𝑧,𝑤)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑈(𝑧,𝑤)   𝐺(𝑧,𝑤)   𝑍(𝑦,𝑧,𝑤)

Proof of Theorem wemapwe
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wemapwe.u . . . . . . . . 9 𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}
2 eqid 2620 . . . . . . . . 9 {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)}
3 eqid 2620 . . . . . . . . 9 (𝐺𝑍) = (𝐺𝑍)
4 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐴 ∈ V)
5 wemapwe.2 . . . . . . . . . . . 12 (𝜑𝑅 We 𝐴)
65adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑅 We 𝐴)
7 wemapwe.5 . . . . . . . . . . . 12 𝐹 = OrdIso(𝑅, 𝐴)
87oiiso 8427 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
94, 6, 8syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
10 isof1o 6558 . . . . . . . . . 10 (𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) → 𝐹:dom 𝐹1-1-onto𝐴)
119, 10syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹:dom 𝐹1-1-onto𝐴)
12 simprl 793 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐵 ∈ V)
13 wemapwe.3 . . . . . . . . . . . 12 (𝜑𝑆 We 𝐵)
1413adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑆 We 𝐵)
15 wemapwe.6 . . . . . . . . . . . 12 𝐺 = OrdIso(𝑆, 𝐵)
1615oiiso 8427 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝑆 We 𝐵) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
1712, 14, 16syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
18 isof1o 6558 . . . . . . . . . 10 (𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵) → 𝐺:dom 𝐺1-1-onto𝐵)
19 f1ocnv 6136 . . . . . . . . . 10 (𝐺:dom 𝐺1-1-onto𝐵𝐺:𝐵1-1-onto→dom 𝐺)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺:𝐵1-1-onto→dom 𝐺)
217oiexg 8425 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐹 ∈ V)
2221ad2antll 764 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹 ∈ V)
23 dmexg 7082 . . . . . . . . . 10 (𝐹 ∈ V → dom 𝐹 ∈ V)
2422, 23syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐹 ∈ V)
2515oiexg 8425 . . . . . . . . . . 11 (𝐵 ∈ V → 𝐺 ∈ V)
2625ad2antrl 763 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺 ∈ V)
27 dmexg 7082 . . . . . . . . . 10 (𝐺 ∈ V → dom 𝐺 ∈ V)
2826, 27syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ∈ V)
29 wemapwe.7 . . . . . . . . . 10 𝑍 = (𝐺‘∅)
3017, 18syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺:dom 𝐺1-1-onto𝐵)
31 f1ofo 6131 . . . . . . . . . . . . . . 15 (𝐺:dom 𝐺1-1-onto𝐵𝐺:dom 𝐺onto𝐵)
32 forn 6105 . . . . . . . . . . . . . . 15 (𝐺:dom 𝐺onto𝐵 → ran 𝐺 = 𝐵)
3330, 31, 323syl 18 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ran 𝐺 = 𝐵)
34 wemapwe.4 . . . . . . . . . . . . . . 15 (𝜑𝐵 ≠ ∅)
3534adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐵 ≠ ∅)
3633, 35eqnetrd 2858 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ran 𝐺 ≠ ∅)
37 dm0rn0 5331 . . . . . . . . . . . . . 14 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
3837necon3bii 2843 . . . . . . . . . . . . 13 (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
3936, 38sylibr 224 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ≠ ∅)
4015oicl 8419 . . . . . . . . . . . . 13 Ord dom 𝐺
41 ord0eln0 5767 . . . . . . . . . . . . 13 (Ord dom 𝐺 → (∅ ∈ dom 𝐺 ↔ dom 𝐺 ≠ ∅))
4240, 41ax-mp 5 . . . . . . . . . . . 12 (∅ ∈ dom 𝐺 ↔ dom 𝐺 ≠ ∅)
4339, 42sylibr 224 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ∅ ∈ dom 𝐺)
4415oif 8420 . . . . . . . . . . . 12 𝐺:dom 𝐺𝐵
4544ffvelrni 6344 . . . . . . . . . . 11 (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ 𝐵)
4643, 45syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺‘∅) ∈ 𝐵)
4729, 46syl5eqel 2703 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑍𝐵)
481, 2, 3, 11, 20, 4, 12, 24, 28, 47mapfien 8298 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→{𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)})
49 eqid 2620 . . . . . . . . . . 11 {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅}
5015oion 8426 . . . . . . . . . . . 12 (𝐵 ∈ V → dom 𝐺 ∈ On)
5150ad2antrl 763 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ∈ On)
527oion 8426 . . . . . . . . . . . 12 (𝐴 ∈ V → dom 𝐹 ∈ On)
5352ad2antll 764 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐹 ∈ On)
5449, 51, 53cantnfdm 8546 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅})
5529fveq2i 6181 . . . . . . . . . . . . 13 (𝐺𝑍) = (𝐺‘(𝐺‘∅))
56 f1ocnvfv1 6517 . . . . . . . . . . . . . 14 ((𝐺:dom 𝐺1-1-onto𝐵 ∧ ∅ ∈ dom 𝐺) → (𝐺‘(𝐺‘∅)) = ∅)
5730, 43, 56syl2anc 692 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺‘(𝐺‘∅)) = ∅)
5855, 57syl5eq 2666 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺𝑍) = ∅)
5958breq2d 4656 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑥 finSupp (𝐺𝑍) ↔ 𝑥 finSupp ∅))
6059rabbidv 3184 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅})
6154, 60eqtr4d 2657 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)})
62 f1oeq3 6116 . . . . . . . . 9 (dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹) ↔ (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→{𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)}))
6361, 62syl 17 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹) ↔ (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→{𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)}))
6448, 63mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹))
65 eqid 2620 . . . . . . . . 9 dom (dom 𝐺 CNF dom 𝐹) = dom (dom 𝐺 CNF dom 𝐹)
66 eqid 2620 . . . . . . . . 9 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))}
6765, 51, 53, 66oemapwe 8576 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ({⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} We dom (dom 𝐺 CNF dom 𝐹) ∧ dom OrdIso({⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))}, dom (dom 𝐺 CNF dom 𝐹)) = (dom 𝐺𝑜 dom 𝐹)))
6867simpld 475 . . . . . . 7 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} We dom (dom 𝐺 CNF dom 𝐹))
69 eqid 2620 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)}
7069f1owe 6588 . . . . . . 7 ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹) → ({⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} We dom (dom 𝐺 CNF dom 𝐹) → {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} We 𝑈))
7164, 68, 70sylc 65 . . . . . 6 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} We 𝑈)
72 weinxp 5176 . . . . . 6 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} We 𝑈 ↔ ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈)
7371, 72sylib 208 . . . . 5 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈)
7411adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐹:dom 𝐹1-1-onto𝐴)
75 f1ofn 6125 . . . . . . . . . . . 12 (𝐹:dom 𝐹1-1-onto𝐴𝐹 Fn dom 𝐹)
76 fveq2 6178 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → (𝑥𝑧) = (𝑥‘(𝐹𝑐)))
77 fveq2 6178 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → (𝑦𝑧) = (𝑦‘(𝐹𝑐)))
7876, 77breq12d 4657 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑐) → ((𝑥𝑧)𝑆(𝑦𝑧) ↔ (𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐))))
79 breq1 4647 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑐) → (𝑧𝑅𝑤 ↔ (𝐹𝑐)𝑅𝑤))
8079imbi1d 331 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → ((𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
8180ralbidv 2983 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑐) → (∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
8278, 81anbi12d 746 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑐) → (((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
8382rexrn 6347 . . . . . . . . . . . 12 (𝐹 Fn dom 𝐹 → (∃𝑧 ∈ ran 𝐹((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
8474, 75, 833syl 18 . . . . . . . . . . 11 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑧 ∈ ran 𝐹((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
85 f1ofo 6131 . . . . . . . . . . . . 13 (𝐹:dom 𝐹1-1-onto𝐴𝐹:dom 𝐹onto𝐴)
86 forn 6105 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐴 → ran 𝐹 = 𝐴)
8774, 85, 863syl 18 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ran 𝐹 = 𝐴)
8887rexeqdv 3140 . . . . . . . . . . 11 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑧 ∈ ran 𝐹((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
8926adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐺 ∈ V)
90 cnvexg 7097 . . . . . . . . . . . . . . 15 (𝐺 ∈ V → 𝐺 ∈ V)
9189, 90syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐺 ∈ V)
92 vex 3198 . . . . . . . . . . . . . . 15 𝑥 ∈ V
9322adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐹 ∈ V)
94 coexg 7102 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝐹 ∈ V) → (𝑥𝐹) ∈ V)
9592, 93, 94sylancr 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝑥𝐹) ∈ V)
96 coexg 7102 . . . . . . . . . . . . . 14 ((𝐺 ∈ V ∧ (𝑥𝐹) ∈ V) → (𝐺 ∘ (𝑥𝐹)) ∈ V)
9791, 95, 96syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝐺 ∘ (𝑥𝐹)) ∈ V)
98 vex 3198 . . . . . . . . . . . . . . 15 𝑦 ∈ V
99 coexg 7102 . . . . . . . . . . . . . . 15 ((𝑦 ∈ V ∧ 𝐹 ∈ V) → (𝑦𝐹) ∈ V)
10098, 93, 99sylancr 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝑦𝐹) ∈ V)
101 coexg 7102 . . . . . . . . . . . . . 14 ((𝐺 ∈ V ∧ (𝑦𝐹) ∈ V) → (𝐺 ∘ (𝑦𝐹)) ∈ V)
10291, 100, 101syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝐺 ∘ (𝑦𝐹)) ∈ V)
103 fveq1 6177 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝐺 ∘ (𝑥𝐹)) → (𝑎𝑐) = ((𝐺 ∘ (𝑥𝐹))‘𝑐))
104 fveq1 6177 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝐺 ∘ (𝑦𝐹)) → (𝑏𝑐) = ((𝐺 ∘ (𝑦𝐹))‘𝑐))
105 eleq12 2689 . . . . . . . . . . . . . . . . 17 (((𝑎𝑐) = ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∧ (𝑏𝑐) = ((𝐺 ∘ (𝑦𝐹))‘𝑐)) → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
106103, 104, 105syl2an 494 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
107 fveq1 6177 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝐺 ∘ (𝑥𝐹)) → (𝑎𝑑) = ((𝐺 ∘ (𝑥𝐹))‘𝑑))
108 fveq1 6177 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝐺 ∘ (𝑦𝐹)) → (𝑏𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))
109107, 108eqeqan12d 2636 . . . . . . . . . . . . . . . . . 18 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑎𝑑) = (𝑏𝑑) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))
110109imbi2d 330 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)) ↔ (𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
111110ralbidv 2983 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → (∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)) ↔ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
112106, 111anbi12d 746 . . . . . . . . . . . . . . 15 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → (((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑))) ↔ (((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
113112rexbidv 3048 . . . . . . . . . . . . . 14 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → (∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑))) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
114113, 66brabga 4979 . . . . . . . . . . . . 13 (((𝐺 ∘ (𝑥𝐹)) ∈ V ∧ (𝐺 ∘ (𝑦𝐹)) ∈ V) → ((𝐺 ∘ (𝑥𝐹)){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} (𝐺 ∘ (𝑦𝐹)) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
11597, 102, 114syl2anc 692 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝐺 ∘ (𝑥𝐹)){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} (𝐺 ∘ (𝑦𝐹)) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
116 simprl 793 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥𝑈)
117 coeq1 5268 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑥 → (𝑓𝐹) = (𝑥𝐹))
118117coeq2d 5273 . . . . . . . . . . . . . . 15 (𝑓 = 𝑥 → (𝐺 ∘ (𝑓𝐹)) = (𝐺 ∘ (𝑥𝐹)))
119 eqid 2620 . . . . . . . . . . . . . . 15 (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))) = (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))
120118, 119fvmptg 6267 . . . . . . . . . . . . . 14 ((𝑥𝑈 ∧ (𝐺 ∘ (𝑥𝐹)) ∈ V) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥) = (𝐺 ∘ (𝑥𝐹)))
121116, 97, 120syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥) = (𝐺 ∘ (𝑥𝐹)))
122 simprr 795 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦𝑈)
123 coeq1 5268 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑦 → (𝑓𝐹) = (𝑦𝐹))
124123coeq2d 5273 . . . . . . . . . . . . . . 15 (𝑓 = 𝑦 → (𝐺 ∘ (𝑓𝐹)) = (𝐺 ∘ (𝑦𝐹)))
125124, 119fvmptg 6267 . . . . . . . . . . . . . 14 ((𝑦𝑈 ∧ (𝐺 ∘ (𝑦𝐹)) ∈ V) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) = (𝐺 ∘ (𝑦𝐹)))
126122, 102, 125syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) = (𝐺 ∘ (𝑦𝐹)))
127121, 126breq12d 4657 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ↔ (𝐺 ∘ (𝑥𝐹)){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} (𝐺 ∘ (𝑦𝐹))))
12817ad2antrr 761 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
129 isocnv 6565 . . . . . . . . . . . . . . . . . 18 (𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵) → 𝐺 Isom 𝑆, E (𝐵, dom 𝐺))
130128, 129syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝐺 Isom 𝑆, E (𝐵, dom 𝐺))
131 ssrab2 3679 . . . . . . . . . . . . . . . . . . . . 21 {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} ⊆ (𝐵𝑚 𝐴)
1321, 131eqsstri 3627 . . . . . . . . . . . . . . . . . . . 20 𝑈 ⊆ (𝐵𝑚 𝐴)
133132, 116sseldi 3593 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥 ∈ (𝐵𝑚 𝐴))
134 elmapi 7864 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐵𝑚 𝐴) → 𝑥:𝐴𝐵)
135133, 134syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥:𝐴𝐵)
1367oif 8420 . . . . . . . . . . . . . . . . . . 19 𝐹:dom 𝐹𝐴
137136ffvelrni 6344 . . . . . . . . . . . . . . . . . 18 (𝑐 ∈ dom 𝐹 → (𝐹𝑐) ∈ 𝐴)
138 ffvelrn 6343 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴𝐵 ∧ (𝐹𝑐) ∈ 𝐴) → (𝑥‘(𝐹𝑐)) ∈ 𝐵)
139135, 137, 138syl2an 494 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑥‘(𝐹𝑐)) ∈ 𝐵)
140132, 122sseldi 3593 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦 ∈ (𝐵𝑚 𝐴))
141 elmapi 7864 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐵𝑚 𝐴) → 𝑦:𝐴𝐵)
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦:𝐴𝐵)
143 ffvelrn 6343 . . . . . . . . . . . . . . . . . 18 ((𝑦:𝐴𝐵 ∧ (𝐹𝑐) ∈ 𝐴) → (𝑦‘(𝐹𝑐)) ∈ 𝐵)
144142, 137, 143syl2an 494 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑦‘(𝐹𝑐)) ∈ 𝐵)
145 isorel 6561 . . . . . . . . . . . . . . . . 17 ((𝐺 Isom 𝑆, E (𝐵, dom 𝐺) ∧ ((𝑥‘(𝐹𝑐)) ∈ 𝐵 ∧ (𝑦‘(𝐹𝑐)) ∈ 𝐵)) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐)))))
146130, 139, 144, 145syl12anc 1322 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐)))))
147 fvex 6188 . . . . . . . . . . . . . . . . 17 (𝐺‘(𝑦‘(𝐹𝑐))) ∈ V
148147epelc 5021 . . . . . . . . . . . . . . . 16 ((𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐))) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐))))
149146, 148syl6bb 276 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐)))))
150135adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑥:𝐴𝐵)
151 fco 6045 . . . . . . . . . . . . . . . . . . 19 ((𝑥:𝐴𝐵𝐹:dom 𝐹𝐴) → (𝑥𝐹):dom 𝐹𝐵)
152150, 136, 151sylancl 693 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑥𝐹):dom 𝐹𝐵)
153 fvco3 6262 . . . . . . . . . . . . . . . . . 18 (((𝑥𝐹):dom 𝐹𝐵𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘((𝑥𝐹)‘𝑐)))
154152, 153sylancom 700 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘((𝑥𝐹)‘𝑐)))
155 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑐 ∈ dom 𝐹)
156 fvco3 6262 . . . . . . . . . . . . . . . . . . 19 ((𝐹:dom 𝐹𝐴𝑐 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑐) = (𝑥‘(𝐹𝑐)))
157136, 155, 156sylancr 694 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑐) = (𝑥‘(𝐹𝑐)))
158157fveq2d 6182 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝐺‘((𝑥𝐹)‘𝑐)) = (𝐺‘(𝑥‘(𝐹𝑐))))
159154, 158eqtrd 2654 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘(𝑥‘(𝐹𝑐))))
160142adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑦:𝐴𝐵)
161 fco 6045 . . . . . . . . . . . . . . . . . . 19 ((𝑦:𝐴𝐵𝐹:dom 𝐹𝐴) → (𝑦𝐹):dom 𝐹𝐵)
162160, 136, 161sylancl 693 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑦𝐹):dom 𝐹𝐵)
163 fvco3 6262 . . . . . . . . . . . . . . . . . 18 (((𝑦𝐹):dom 𝐹𝐵𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘((𝑦𝐹)‘𝑐)))
164162, 163sylancom 700 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘((𝑦𝐹)‘𝑐)))
165 fvco3 6262 . . . . . . . . . . . . . . . . . . 19 ((𝐹:dom 𝐹𝐴𝑐 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑐) = (𝑦‘(𝐹𝑐)))
166136, 155, 165sylancr 694 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑐) = (𝑦‘(𝐹𝑐)))
167166fveq2d 6182 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝐺‘((𝑦𝐹)‘𝑐)) = (𝐺‘(𝑦‘(𝐹𝑐))))
168164, 167eqtrd 2654 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘(𝑦‘(𝐹𝑐))))
169159, 168eleq12d 2693 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐)))))
170149, 169bitr4d 271 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
17187raleqdv 3139 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
172 breq2 4648 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹𝑑) → ((𝐹𝑐)𝑅𝑤 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
173 fveq2 6178 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹𝑑) → (𝑥𝑤) = (𝑥‘(𝐹𝑑)))
174 fveq2 6178 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹𝑑) → (𝑦𝑤) = (𝑦‘(𝐹𝑑)))
175173, 174eqeq12d 2635 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹𝑑) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
176172, 175imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹𝑑) → (((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
177176ralrn 6348 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn dom 𝐹 → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
17874, 75, 1773syl 18 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
179171, 178bitr3d 270 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
180179adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
181 epel 5022 . . . . . . . . . . . . . . . . . . 19 (𝑐 E 𝑑𝑐𝑑)
1829ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
183 isorel 6561 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐 E 𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
184182, 183sylancom 700 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐 E 𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
185181, 184syl5bbr 274 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
186152adantrr 752 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑥𝐹):dom 𝐹𝐵)
187 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝑑 ∈ dom 𝐹)
188 fvco3 6262 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝐹):dom 𝐹𝐵𝑑 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = (𝐺‘((𝑥𝐹)‘𝑑)))
189186, 187, 188syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = (𝐺‘((𝑥𝐹)‘𝑑)))
190162adantrr 752 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑦𝐹):dom 𝐹𝐵)
191 fvco3 6262 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦𝐹):dom 𝐹𝐵𝑑 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑑) = (𝐺‘((𝑦𝐹)‘𝑑)))
192190, 187, 191syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺 ∘ (𝑦𝐹))‘𝑑) = (𝐺‘((𝑦𝐹)‘𝑑)))
193189, 192eqeq12d 2635 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑) ↔ (𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑))))
19430ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐺:dom 𝐺1-1-onto𝐵)
195 f1of1 6123 . . . . . . . . . . . . . . . . . . . . 21 (𝐺:𝐵1-1-onto→dom 𝐺𝐺:𝐵1-1→dom 𝐺)
196194, 19, 1953syl 18 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐺:𝐵1-1→dom 𝐺)
197186, 187ffvelrnd 6346 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑥𝐹)‘𝑑) ∈ 𝐵)
198190, 187ffvelrnd 6346 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑦𝐹)‘𝑑) ∈ 𝐵)
199 f1fveq 6504 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:𝐵1-1→dom 𝐺 ∧ (((𝑥𝐹)‘𝑑) ∈ 𝐵 ∧ ((𝑦𝐹)‘𝑑) ∈ 𝐵)) → ((𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑)) ↔ ((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑)))
200196, 197, 198, 199syl12anc 1322 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑)) ↔ ((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑)))
201 fvco3 6262 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:dom 𝐹𝐴𝑑 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑑) = (𝑥‘(𝐹𝑑)))
202136, 187, 201sylancr 694 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑥𝐹)‘𝑑) = (𝑥‘(𝐹𝑑)))
203 fvco3 6262 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:dom 𝐹𝐴𝑑 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑑) = (𝑦‘(𝐹𝑑)))
204136, 187, 203sylancr 694 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑦𝐹)‘𝑑) = (𝑦‘(𝐹𝑑)))
205202, 204eqeq12d 2635 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
206193, 200, 2053bitrd 294 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
207185, 206imbi12d 334 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)) ↔ ((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
208207anassrs 679 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) ∧ 𝑑 ∈ dom 𝐹) → ((𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)) ↔ ((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
209208ralbidva 2982 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
210180, 209bitr4d 271 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
211170, 210anbi12d 746 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ (((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
212211rexbidva 3045 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
213115, 127, 2123bitr4rd 301 . . . . . . . . . . 11 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)))
21484, 88, 2133bitr3d 298 . . . . . . . . . 10 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)))
215214ex 450 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((𝑥𝑈𝑦𝑈) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦))))
216215pm5.32rd 671 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈)) ↔ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))))
217216opabbidv 4707 . . . . . . 7 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈))} = {⟨𝑥, 𝑦⟩ ∣ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))})
218 wemapwe.t . . . . . . . . 9 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
219 df-xp 5110 . . . . . . . . 9 (𝑈 × 𝑈) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)}
220218, 219ineq12i 3804 . . . . . . . 8 (𝑇 ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)})
221 inopab 5241 . . . . . . . 8 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈))}
222220, 221eqtri 2642 . . . . . . 7 (𝑇 ∩ (𝑈 × 𝑈)) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈))}
223219ineq2i 3803 . . . . . . . 8 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)})
224 inopab 5241 . . . . . . . 8 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)}) = {⟨𝑥, 𝑦⟩ ∣ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))}
225223, 224eqtri 2642 . . . . . . 7 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) = {⟨𝑥, 𝑦⟩ ∣ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))}
226217, 222, 2253eqtr4g 2679 . . . . . 6 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑇 ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)))
227 weeq1 5092 . . . . . 6 ((𝑇 ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) → ((𝑇 ∩ (𝑈 × 𝑈)) We 𝑈 ↔ ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈))
228226, 227syl 17 . . . . 5 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((𝑇 ∩ (𝑈 × 𝑈)) We 𝑈 ↔ ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈))
22973, 228mpbird 247 . . . 4 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑇 ∩ (𝑈 × 𝑈)) We 𝑈)
230 weinxp 5176 . . . 4 (𝑇 We 𝑈 ↔ (𝑇 ∩ (𝑈 × 𝑈)) We 𝑈)
231229, 230sylibr 224 . . 3 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑇 We 𝑈)
232231ex 450 . 2 (𝜑 → ((𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑇 We 𝑈))
233 we0 5099 . . 3 𝑇 We ∅
234 elmapex 7863 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑚 𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
235234con3i 150 . . . . . . . 8 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ¬ 𝑥 ∈ (𝐵𝑚 𝐴))
236235pm2.21d 118 . . . . . . 7 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐵𝑚 𝐴) → ¬ 𝑥 finSupp 𝑍))
237236ralrimiv 2962 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ∀𝑥 ∈ (𝐵𝑚 𝐴) ¬ 𝑥 finSupp 𝑍)
238 rabeq0 3948 . . . . . 6 ({𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅ ↔ ∀𝑥 ∈ (𝐵𝑚 𝐴) ¬ 𝑥 finSupp 𝑍)
239237, 238sylibr 224 . . . . 5 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅)
2401, 239syl5eq 2666 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑈 = ∅)
241 weeq2 5093 . . . 4 (𝑈 = ∅ → (𝑇 We 𝑈𝑇 We ∅))
242240, 241syl 17 . . 3 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝑇 We 𝑈𝑇 We ∅))
243233, 242mpbiri 248 . 2 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑇 We 𝑈)
244232, 243pm2.61d1 171 1 (𝜑𝑇 We 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  cin 3566  c0 3907   class class class wbr 4644  {copab 4703  cmpt 4720   E cep 5018   We wwe 5062   × cxp 5102  ccnv 5103  dom cdm 5104  ran crn 5105  ccom 5108  Ord word 5710  Oncon0 5711   Fn wfn 5871  wf 5872  1-1wf1 5873  ontowfo 5874  1-1-ontowf1o 5875  cfv 5876   Isom wiso 5877  (class class class)co 6635  𝑜 coe 7544  𝑚 cmap 7842   finSupp cfsupp 8260  OrdIsocoi 8399   CNF ccnf 8543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-2o 7546  df-oadd 7549  df-omul 7550  df-oexp 7551  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-oi 8400  df-cnf 8544
This theorem is referenced by:  ltbwe  19453
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