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Theorem fpwwe2lem9 9661
Description: Lemma for fpwwe2 9666. Given two well-orders 𝑋, 𝑅 and 𝑌, 𝑆 of parts of 𝐴, one is an initial segment of the other. (The 𝑂𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2lem9.x (𝜑𝑋𝑊𝑅)
fpwwe2lem9.y (𝜑𝑌𝑊𝑆)
fpwwe2lem9.m 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem9.n 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem9.s (𝜑 → dom 𝑀 ⊆ dom 𝑁)
Assertion
Ref Expression
fpwwe2lem9 (𝜑 → (𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝑀,𝑟,𝑢,𝑥,𝑦   𝑁,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑆,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)

Proof of Theorem fpwwe2lem9
StepHypRef Expression
1 fpwwe2lem9.x . . . . . . . . 9 (𝜑𝑋𝑊𝑅)
2 fpwwe2.1 . . . . . . . . . . 11 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
32relopabi 5384 . . . . . . . . . 10 Rel 𝑊
43brrelexi 5298 . . . . . . . . 9 (𝑋𝑊𝑅𝑋 ∈ V)
51, 4syl 17 . . . . . . . 8 (𝜑𝑋 ∈ V)
6 fpwwe2.2 . . . . . . . . . . . 12 (𝜑𝐴 ∈ V)
72, 6fpwwe2lem2 9655 . . . . . . . . . . 11 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
81, 7mpbid 222 . . . . . . . . . 10 (𝜑 → ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
98simprd 477 . . . . . . . . 9 (𝜑 → (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
109simpld 476 . . . . . . . 8 (𝜑𝑅 We 𝑋)
11 fpwwe2lem9.m . . . . . . . . 9 𝑀 = OrdIso(𝑅, 𝑋)
1211oiiso 8597 . . . . . . . 8 ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
135, 10, 12syl2anc 565 . . . . . . 7 (𝜑𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
14 isof1o 6715 . . . . . . 7 (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀1-1-onto𝑋)
1513, 14syl 17 . . . . . 6 (𝜑𝑀:dom 𝑀1-1-onto𝑋)
16 f1ofo 6285 . . . . . 6 (𝑀:dom 𝑀1-1-onto𝑋𝑀:dom 𝑀onto𝑋)
17 forn 6259 . . . . . 6 (𝑀:dom 𝑀onto𝑋 → ran 𝑀 = 𝑋)
1815, 16, 173syl 18 . . . . 5 (𝜑 → ran 𝑀 = 𝑋)
19 fpwwe2.3 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
20 fpwwe2lem9.y . . . . . . 7 (𝜑𝑌𝑊𝑆)
21 fpwwe2lem9.n . . . . . . 7 𝑁 = OrdIso(𝑆, 𝑌)
22 fpwwe2lem9.s . . . . . . 7 (𝜑 → dom 𝑀 ⊆ dom 𝑁)
232, 6, 19, 1, 20, 11, 21, 22fpwwe2lem8 9660 . . . . . 6 (𝜑𝑀 = (𝑁 ↾ dom 𝑀))
2423rneqd 5491 . . . . 5 (𝜑 → ran 𝑀 = ran (𝑁 ↾ dom 𝑀))
2518, 24eqtr3d 2806 . . . 4 (𝜑𝑋 = ran (𝑁 ↾ dom 𝑀))
26 df-ima 5262 . . . 4 (𝑁 “ dom 𝑀) = ran (𝑁 ↾ dom 𝑀)
2725, 26syl6eqr 2822 . . 3 (𝜑𝑋 = (𝑁 “ dom 𝑀))
28 imassrn 5618 . . . 4 (𝑁 “ dom 𝑀) ⊆ ran 𝑁
293brrelexi 5298 . . . . . . . 8 (𝑌𝑊𝑆𝑌 ∈ V)
3020, 29syl 17 . . . . . . 7 (𝜑𝑌 ∈ V)
312, 6fpwwe2lem2 9655 . . . . . . . . . 10 (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌𝐴𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
3220, 31mpbid 222 . . . . . . . . 9 (𝜑 → ((𝑌𝐴𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
3332simprd 477 . . . . . . . 8 (𝜑 → (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))
3433simpld 476 . . . . . . 7 (𝜑𝑆 We 𝑌)
3521oiiso 8597 . . . . . . 7 ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
3630, 34, 35syl2anc 565 . . . . . 6 (𝜑𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
37 isof1o 6715 . . . . . 6 (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁1-1-onto𝑌)
3836, 37syl 17 . . . . 5 (𝜑𝑁:dom 𝑁1-1-onto𝑌)
39 f1ofo 6285 . . . . 5 (𝑁:dom 𝑁1-1-onto𝑌𝑁:dom 𝑁onto𝑌)
40 forn 6259 . . . . 5 (𝑁:dom 𝑁onto𝑌 → ran 𝑁 = 𝑌)
4138, 39, 403syl 18 . . . 4 (𝜑 → ran 𝑁 = 𝑌)
4228, 41syl5sseq 3800 . . 3 (𝜑 → (𝑁 “ dom 𝑀) ⊆ 𝑌)
4327, 42eqsstrd 3786 . 2 (𝜑𝑋𝑌)
448simpld 476 . . . . . 6 (𝜑 → (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)))
4544simprd 477 . . . . 5 (𝜑𝑅 ⊆ (𝑋 × 𝑋))
46 relxp 5266 . . . . 5 Rel (𝑋 × 𝑋)
47 relss 5346 . . . . 5 (𝑅 ⊆ (𝑋 × 𝑋) → (Rel (𝑋 × 𝑋) → Rel 𝑅))
4845, 46, 47mpisyl 21 . . . 4 (𝜑 → Rel 𝑅)
49 inss2 3980 . . . . 5 (𝑆 ∩ (𝑌 × 𝑋)) ⊆ (𝑌 × 𝑋)
50 relxp 5266 . . . . 5 Rel (𝑌 × 𝑋)
51 relss 5346 . . . . 5 ((𝑆 ∩ (𝑌 × 𝑋)) ⊆ (𝑌 × 𝑋) → (Rel (𝑌 × 𝑋) → Rel (𝑆 ∩ (𝑌 × 𝑋))))
5249, 50, 51mp2 9 . . . 4 Rel (𝑆 ∩ (𝑌 × 𝑋))
5348, 52jctir 504 . . 3 (𝜑 → (Rel 𝑅 ∧ Rel (𝑆 ∩ (𝑌 × 𝑋))))
5445ssbrd 4827 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦𝑥(𝑋 × 𝑋)𝑦))
55 brxp 5287 . . . . . . 7 (𝑥(𝑋 × 𝑋)𝑦 ↔ (𝑥𝑋𝑦𝑋))
5654, 55syl6ib 241 . . . . . 6 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝑋𝑦𝑋)))
57 brinxp2 5320 . . . . . . . 8 (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ (𝑥𝑌𝑦𝑋𝑥𝑆𝑦))
58 df-3an 1072 . . . . . . . 8 ((𝑥𝑌𝑦𝑋𝑥𝑆𝑦) ↔ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦))
5957, 58bitri 264 . . . . . . 7 (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦))
60 simprll 756 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥𝑌)
61 simprr 748 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥𝑆𝑦)
62 isocnv 6722 . . . . . . . . . . . . . . . . 17 (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
6336, 62syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
6463adantr 466 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
6543adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋𝑌)
66 simprlr 757 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦𝑋)
6765, 66sseldd 3751 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦𝑌)
68 isorel 6718 . . . . . . . . . . . . . . 15 ((𝑁 Isom 𝑆, E (𝑌, dom 𝑁) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑆𝑦 ↔ (𝑁𝑥) E (𝑁𝑦)))
6964, 60, 67, 68syl12anc 1473 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥𝑆𝑦 ↔ (𝑁𝑥) E (𝑁𝑦)))
7061, 69mpbid 222 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑁𝑥) E (𝑁𝑦))
71 fvex 6342 . . . . . . . . . . . . . 14 (𝑁𝑦) ∈ V
7271epelc 5164 . . . . . . . . . . . . 13 ((𝑁𝑥) E (𝑁𝑦) ↔ (𝑁𝑥) ∈ (𝑁𝑦))
7370, 72sylib 208 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑁𝑥) ∈ (𝑁𝑦))
7423adantr 466 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑀 = (𝑁 ↾ dom 𝑀))
7574cnveqd 5436 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑀 = (𝑁 ↾ dom 𝑀))
76 isof1o 6715 . . . . . . . . . . . . . . . . . 18 (𝑁 Isom 𝑆, E (𝑌, dom 𝑁) → 𝑁:𝑌1-1-onto→dom 𝑁)
77 f1ofn 6279 . . . . . . . . . . . . . . . . . 18 (𝑁:𝑌1-1-onto→dom 𝑁𝑁 Fn 𝑌)
7864, 76, 773syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑁 Fn 𝑌)
79 fnfun 6128 . . . . . . . . . . . . . . . . 17 (𝑁 Fn 𝑌 → Fun 𝑁)
80 funcnvres 6107 . . . . . . . . . . . . . . . . 17 (Fun 𝑁(𝑁 ↾ dom 𝑀) = (𝑁 ↾ (𝑁 “ dom 𝑀)))
8178, 79, 803syl 18 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑁 ↾ dom 𝑀) = (𝑁 ↾ (𝑁 “ dom 𝑀)))
8275, 81eqtrd 2804 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑀 = (𝑁 ↾ (𝑁 “ dom 𝑀)))
8382fveq1d 6334 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑀𝑦) = ((𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦))
8427adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 = (𝑁 “ dom 𝑀))
8566, 84eleqtrd 2851 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦 ∈ (𝑁 “ dom 𝑀))
86 fvres 6348 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑁 “ dom 𝑀) → ((𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦) = (𝑁𝑦))
8785, 86syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → ((𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦) = (𝑁𝑦))
8883, 87eqtrd 2804 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑀𝑦) = (𝑁𝑦))
89 isocnv 6722 . . . . . . . . . . . . . . . . 17 (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
9013, 89syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
91 isof1o 6715 . . . . . . . . . . . . . . . 16 (𝑀 Isom 𝑅, E (𝑋, dom 𝑀) → 𝑀:𝑋1-1-onto→dom 𝑀)
92 f1of 6278 . . . . . . . . . . . . . . . 16 (𝑀:𝑋1-1-onto→dom 𝑀𝑀:𝑋⟶dom 𝑀)
9390, 91, 923syl 18 . . . . . . . . . . . . . . 15 (𝜑𝑀:𝑋⟶dom 𝑀)
9493adantr 466 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑀:𝑋⟶dom 𝑀)
9594, 66ffvelrnd 6503 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑀𝑦) ∈ dom 𝑀)
9688, 95eqeltrrd 2850 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑁𝑦) ∈ dom 𝑀)
9711oicl 8589 . . . . . . . . . . . . 13 Ord dom 𝑀
98 ordtr1 5910 . . . . . . . . . . . . 13 (Ord dom 𝑀 → (((𝑁𝑥) ∈ (𝑁𝑦) ∧ (𝑁𝑦) ∈ dom 𝑀) → (𝑁𝑥) ∈ dom 𝑀))
9997, 98ax-mp 5 . . . . . . . . . . . 12 (((𝑁𝑥) ∈ (𝑁𝑦) ∧ (𝑁𝑦) ∈ dom 𝑀) → (𝑁𝑥) ∈ dom 𝑀)
10073, 96, 99syl2anc 565 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑁𝑥) ∈ dom 𝑀)
101 elpreima 6480 . . . . . . . . . . . 12 (𝑁 Fn 𝑌 → (𝑥 ∈ (𝑁 “ dom 𝑀) ↔ (𝑥𝑌 ∧ (𝑁𝑥) ∈ dom 𝑀)))
10278, 101syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥 ∈ (𝑁 “ dom 𝑀) ↔ (𝑥𝑌 ∧ (𝑁𝑥) ∈ dom 𝑀)))
10360, 100, 102mpbir2and 684 . . . . . . . . . 10 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥 ∈ (𝑁 “ dom 𝑀))
104 imacnvcnv 5740 . . . . . . . . . . 11 (𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀)
10584, 104syl6eqr 2822 . . . . . . . . . 10 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 = (𝑁 “ dom 𝑀))
106103, 105eleqtrrd 2852 . . . . . . . . 9 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥𝑋)
107106, 66jca 495 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥𝑋𝑦𝑋))
108107ex 397 . . . . . . 7 (𝜑 → (((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦) → (𝑥𝑋𝑦𝑋)))
10959, 108syl5bi 232 . . . . . 6 (𝜑 → (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 → (𝑥𝑋𝑦𝑋)))
11023adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑀 = (𝑁 ↾ dom 𝑀))
111110cnveqd 5436 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑀 = (𝑁 ↾ dom 𝑀))
112111fveq1d 6334 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑀𝑥) = ((𝑁 ↾ dom 𝑀)‘𝑥))
113111fveq1d 6334 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑀𝑦) = ((𝑁 ↾ dom 𝑀)‘𝑦))
114112, 113breq12d 4797 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝑀𝑥) E (𝑀𝑦) ↔ ((𝑁 ↾ dom 𝑀)‘𝑥) E ((𝑁 ↾ dom 𝑀)‘𝑦)))
115 isorel 6718 . . . . . . . . . 10 ((𝑀 Isom 𝑅, E (𝑋, dom 𝑀) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦 ↔ (𝑀𝑥) E (𝑀𝑦)))
11690, 115sylan 561 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦 ↔ (𝑀𝑥) E (𝑀𝑦)))
117 eqidd 2771 . . . . . . . . . . . . 13 (𝜑 → (𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀))
118 isores3 6727 . . . . . . . . . . . . 13 ((𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) ∧ dom 𝑀 ⊆ dom 𝑁 ∧ (𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀)) → (𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)))
11936, 22, 117, 118syl3anc 1475 . . . . . . . . . . . 12 (𝜑 → (𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)))
120 isocnv 6722 . . . . . . . . . . . 12 ((𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)) → (𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
121119, 120syl 17 . . . . . . . . . . 11 (𝜑(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
122121adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
123 simprl 746 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
12427adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑋 = (𝑁 “ dom 𝑀))
125123, 124eleqtrd 2851 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 ∈ (𝑁 “ dom 𝑀))
126 simprr 748 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
127126, 124eleqtrd 2851 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑦 ∈ (𝑁 “ dom 𝑀))
128 isorel 6718 . . . . . . . . . 10 (((𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀) ∧ (𝑥 ∈ (𝑁 “ dom 𝑀) ∧ 𝑦 ∈ (𝑁 “ dom 𝑀))) → (𝑥𝑆𝑦 ↔ ((𝑁 ↾ dom 𝑀)‘𝑥) E ((𝑁 ↾ dom 𝑀)‘𝑦)))
129122, 125, 127, 128syl12anc 1473 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑆𝑦 ↔ ((𝑁 ↾ dom 𝑀)‘𝑥) E ((𝑁 ↾ dom 𝑀)‘𝑦)))
130114, 116, 1293bitr4d 300 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦𝑥𝑆𝑦))
13143sselda 3750 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → 𝑥𝑌)
132131adantrr 688 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑌)
133132, 126jca 495 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑌𝑦𝑋))
134133biantrurd 516 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑆𝑦 ↔ ((𝑥𝑌𝑦𝑋) ∧ 𝑥𝑆𝑦)))
135134, 59syl6bbr 278 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑆𝑦𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
136130, 135bitrd 268 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑅𝑦𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
137136ex 397 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦)))
13856, 109, 137pm5.21ndd 368 . . . . 5 (𝜑 → (𝑥𝑅𝑦𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
139 df-br 4785 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
140 df-br 4785 . . . . 5 (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑆 ∩ (𝑌 × 𝑋)))
141138, 139, 1403bitr3g 302 . . . 4 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑆 ∩ (𝑌 × 𝑋))))
142141eqrelrdv2 5359 . . 3 (((Rel 𝑅 ∧ Rel (𝑆 ∩ (𝑌 × 𝑋))) ∧ 𝜑) → 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))
14353, 142mpancom 660 . 2 (𝜑𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))
14443, 143jca 495 1 (𝜑 → (𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  [wsbc 3585  cin 3720  wss 3721  {csn 4314  cop 4320   class class class wbr 4784  {copab 4844   E cep 5161   We wwe 5207   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Rel wrel 5254  Ord word 5865  Fun wfun 6025   Fn wfn 6026  wf 6027  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031   Isom wiso 6032  (class class class)co 6792  OrdIsocoi 8569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-wrecs 7558  df-recs 7620  df-oi 8570
This theorem is referenced by:  fpwwe2lem10  9662
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