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Theorem fseqenlem1 9068
Description: Lemma for fseqen 9071. (Contributed by Mario Carneiro, 17-May-2015.)
Hypotheses
Ref Expression
fseqenlem.a (𝜑𝐴𝑉)
fseqenlem.b (𝜑𝐵𝐴)
fseqenlem.f (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
fseqenlem.g 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
Assertion
Ref Expression
fseqenlem1 ((𝜑𝐶 ∈ ω) → (𝐺𝐶):(𝐴𝑚 𝐶)–1-1𝐴)
Distinct variable groups:   𝑓,𝑛,𝑥,𝐹   𝐴,𝑓,𝑛,𝑥   𝜑,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑥,𝑓,𝑛)   𝐶(𝑥,𝑓,𝑛)   𝐺(𝑥,𝑓,𝑛)   𝑉(𝑥,𝑓,𝑛)

Proof of Theorem fseqenlem1
Dummy variables 𝑦 𝑎 𝑏 𝑧 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6348 . . . . . 6 (𝑦 = 𝐶 → (𝐺𝑦) = (𝐺𝐶))
2 f1eq1 6251 . . . . . 6 ((𝐺𝑦) = (𝐺𝐶) → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝐶):(𝐴𝑚 𝑦)–1-1𝐴))
31, 2syl 17 . . . . 5 (𝑦 = 𝐶 → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝐶):(𝐴𝑚 𝑦)–1-1𝐴))
4 oveq2 6820 . . . . . 6 (𝑦 = 𝐶 → (𝐴𝑚 𝑦) = (𝐴𝑚 𝐶))
5 f1eq2 6252 . . . . . 6 ((𝐴𝑚 𝑦) = (𝐴𝑚 𝐶) → ((𝐺𝐶):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝐶):(𝐴𝑚 𝐶)–1-1𝐴))
64, 5syl 17 . . . . 5 (𝑦 = 𝐶 → ((𝐺𝐶):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝐶):(𝐴𝑚 𝐶)–1-1𝐴))
73, 6bitrd 269 . . . 4 (𝑦 = 𝐶 → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝐶):(𝐴𝑚 𝐶)–1-1𝐴))
87imbi2d 330 . . 3 (𝑦 = 𝐶 → ((𝜑 → (𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴) ↔ (𝜑 → (𝐺𝐶):(𝐴𝑚 𝐶)–1-1𝐴)))
9 fveq2 6348 . . . . . . 7 (𝑦 = ∅ → (𝐺𝑦) = (𝐺‘∅))
10 snex 5050 . . . . . . . 8 {⟨∅, 𝐵⟩} ∈ V
11 fseqenlem.g . . . . . . . . 9 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
1211seqom0g 7725 . . . . . . . 8 ({⟨∅, 𝐵⟩} ∈ V → (𝐺‘∅) = {⟨∅, 𝐵⟩})
1310, 12ax-mp 5 . . . . . . 7 (𝐺‘∅) = {⟨∅, 𝐵⟩}
149, 13syl6eq 2824 . . . . . 6 (𝑦 = ∅ → (𝐺𝑦) = {⟨∅, 𝐵⟩})
15 f1eq1 6251 . . . . . 6 ((𝐺𝑦) = {⟨∅, 𝐵⟩} → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴𝑚 𝑦)–1-1𝐴))
1614, 15syl 17 . . . . 5 (𝑦 = ∅ → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴𝑚 𝑦)–1-1𝐴))
17 oveq2 6820 . . . . . 6 (𝑦 = ∅ → (𝐴𝑚 𝑦) = (𝐴𝑚 ∅))
18 f1eq2 6252 . . . . . 6 ((𝐴𝑚 𝑦) = (𝐴𝑚 ∅) → ({⟨∅, 𝐵⟩}:(𝐴𝑚 𝑦)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴𝑚 ∅)–1-1𝐴))
1917, 18syl 17 . . . . 5 (𝑦 = ∅ → ({⟨∅, 𝐵⟩}:(𝐴𝑚 𝑦)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴𝑚 ∅)–1-1𝐴))
2016, 19bitrd 269 . . . 4 (𝑦 = ∅ → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴𝑚 ∅)–1-1𝐴))
21 fveq2 6348 . . . . . 6 (𝑦 = 𝑚 → (𝐺𝑦) = (𝐺𝑚))
22 f1eq1 6251 . . . . . 6 ((𝐺𝑦) = (𝐺𝑚) → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝑚):(𝐴𝑚 𝑦)–1-1𝐴))
2321, 22syl 17 . . . . 5 (𝑦 = 𝑚 → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝑚):(𝐴𝑚 𝑦)–1-1𝐴))
24 oveq2 6820 . . . . . 6 (𝑦 = 𝑚 → (𝐴𝑚 𝑦) = (𝐴𝑚 𝑚))
25 f1eq2 6252 . . . . . 6 ((𝐴𝑚 𝑦) = (𝐴𝑚 𝑚) → ((𝐺𝑚):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴))
2624, 25syl 17 . . . . 5 (𝑦 = 𝑚 → ((𝐺𝑚):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴))
2723, 26bitrd 269 . . . 4 (𝑦 = 𝑚 → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴))
28 fveq2 6348 . . . . . 6 (𝑦 = suc 𝑚 → (𝐺𝑦) = (𝐺‘suc 𝑚))
29 f1eq1 6251 . . . . . 6 ((𝐺𝑦) = (𝐺‘suc 𝑚) → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺‘suc 𝑚):(𝐴𝑚 𝑦)–1-1𝐴))
3028, 29syl 17 . . . . 5 (𝑦 = suc 𝑚 → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺‘suc 𝑚):(𝐴𝑚 𝑦)–1-1𝐴))
31 oveq2 6820 . . . . . 6 (𝑦 = suc 𝑚 → (𝐴𝑚 𝑦) = (𝐴𝑚 suc 𝑚))
32 f1eq2 6252 . . . . . 6 ((𝐴𝑚 𝑦) = (𝐴𝑚 suc 𝑚) → ((𝐺‘suc 𝑚):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)–1-1𝐴))
3331, 32syl 17 . . . . 5 (𝑦 = suc 𝑚 → ((𝐺‘suc 𝑚):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)–1-1𝐴))
3430, 33bitrd 269 . . . 4 (𝑦 = suc 𝑚 → ((𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴 ↔ (𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)–1-1𝐴))
35 0ex 4937 . . . . . . . 8 ∅ ∈ V
36 fseqenlem.b . . . . . . . 8 (𝜑𝐵𝐴)
37 f1osng 6333 . . . . . . . 8 ((∅ ∈ V ∧ 𝐵𝐴) → {⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵})
3835, 36, 37sylancr 576 . . . . . . 7 (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵})
39 f1of1 6292 . . . . . . 7 ({⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵} → {⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵})
4038, 39syl 17 . . . . . 6 (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵})
4136snssd 4486 . . . . . 6 (𝜑 → {𝐵} ⊆ 𝐴)
42 f1ss 6261 . . . . . 6 (({⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝐴) → {⟨∅, 𝐵⟩}:{∅}–1-1𝐴)
4340, 41, 42syl2anc 574 . . . . 5 (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1𝐴)
44 fseqenlem.a . . . . . . . 8 (𝜑𝐴𝑉)
45 map0e 8068 . . . . . . . 8 (𝐴𝑉 → (𝐴𝑚 ∅) = 1𝑜)
4644, 45syl 17 . . . . . . 7 (𝜑 → (𝐴𝑚 ∅) = 1𝑜)
47 df1o2 7747 . . . . . . 7 1𝑜 = {∅}
4846, 47syl6eq 2824 . . . . . 6 (𝜑 → (𝐴𝑚 ∅) = {∅})
49 f1eq2 6252 . . . . . 6 ((𝐴𝑚 ∅) = {∅} → ({⟨∅, 𝐵⟩}:(𝐴𝑚 ∅)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:{∅}–1-1𝐴))
5048, 49syl 17 . . . . 5 (𝜑 → ({⟨∅, 𝐵⟩}:(𝐴𝑚 ∅)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:{∅}–1-1𝐴))
5143, 50mpbird 248 . . . 4 (𝜑 → {⟨∅, 𝐵⟩}:(𝐴𝑚 ∅)–1-1𝐴)
52 fseqenlem.f . . . . . . . . . . . 12 (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
5352ad2antrr 706 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → 𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
54 f1of 6293 . . . . . . . . . . 11 (𝐹:(𝐴 × 𝐴)–1-1-onto𝐴𝐹:(𝐴 × 𝐴)⟶𝐴)
5553, 54syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → 𝐹:(𝐴 × 𝐴)⟶𝐴)
56 f1f 6256 . . . . . . . . . . . . 13 ((𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴 → (𝐺𝑚):(𝐴𝑚 𝑚)⟶𝐴)
5756ad2antll 709 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → (𝐺𝑚):(𝐴𝑚 𝑚)⟶𝐴)
5857adantr 467 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → (𝐺𝑚):(𝐴𝑚 𝑚)⟶𝐴)
59 elmapi 8052 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴𝑚 suc 𝑚) → 𝑧:suc 𝑚𝐴)
6059adantl 468 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → 𝑧:suc 𝑚𝐴)
61 sssucid 5956 . . . . . . . . . . . . 13 𝑚 ⊆ suc 𝑚
62 fssres 6225 . . . . . . . . . . . . 13 ((𝑧:suc 𝑚𝐴𝑚 ⊆ suc 𝑚) → (𝑧𝑚):𝑚𝐴)
6360, 61, 62sylancl 575 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → (𝑧𝑚):𝑚𝐴)
6444ad2antrr 706 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → 𝐴𝑉)
65 vex 3358 . . . . . . . . . . . . 13 𝑚 ∈ V
66 elmapg 8043 . . . . . . . . . . . . 13 ((𝐴𝑉𝑚 ∈ V) → ((𝑧𝑚) ∈ (𝐴𝑚 𝑚) ↔ (𝑧𝑚):𝑚𝐴))
6764, 65, 66sylancl 575 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → ((𝑧𝑚) ∈ (𝐴𝑚 𝑚) ↔ (𝑧𝑚):𝑚𝐴))
6863, 67mpbird 248 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → (𝑧𝑚) ∈ (𝐴𝑚 𝑚))
6958, 68ffvelrnd 6520 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → ((𝐺𝑚)‘(𝑧𝑚)) ∈ 𝐴)
7065sucid 5958 . . . . . . . . . . 11 𝑚 ∈ suc 𝑚
71 ffvelrn 6517 . . . . . . . . . . 11 ((𝑧:suc 𝑚𝐴𝑚 ∈ suc 𝑚) → (𝑧𝑚) ∈ 𝐴)
7260, 70, 71sylancl 575 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → (𝑧𝑚) ∈ 𝐴)
7355, 69, 72fovrnd 6974 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴𝑚 suc 𝑚)) → (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)) ∈ 𝐴)
7473fmpttd 6545 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))):(𝐴𝑚 suc 𝑚)⟶𝐴)
7511seqomsuc 7726 . . . . . . . . . . 11 (𝑚 ∈ ω → (𝐺‘suc 𝑚) = (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))))(𝐺𝑚)))
7675ad2antrl 708 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → (𝐺‘suc 𝑚) = (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))))(𝐺𝑚)))
77 fvex 6359 . . . . . . . . . . 11 (𝐺𝑚) ∈ V
78 reseq1 5540 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑥𝑎) = (𝑧𝑎))
7978fveq2d 6352 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑏‘(𝑥𝑎)) = (𝑏‘(𝑧𝑎)))
80 fveq1 6347 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑥𝑎) = (𝑧𝑎))
8179, 80oveq12d 6830 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)) = ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎)))
8281cbvmptv 4897 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))) = (𝑧 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎)))
83 suceq 5944 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑚 → suc 𝑎 = suc 𝑚)
8483adantr 467 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → suc 𝑎 = suc 𝑚)
8584oveq2d 6828 . . . . . . . . . . . . . 14 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝐴𝑚 suc 𝑎) = (𝐴𝑚 suc 𝑚))
86 simpr 472 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → 𝑏 = (𝐺𝑚))
87 reseq2 5541 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑚 → (𝑧𝑎) = (𝑧𝑚))
8887adantr 467 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑧𝑎) = (𝑧𝑚))
8986, 88fveq12d 6355 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑏‘(𝑧𝑎)) = ((𝐺𝑚)‘(𝑧𝑚)))
90 simpl 469 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → 𝑎 = 𝑚)
9190fveq2d 6352 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑧𝑎) = (𝑧𝑚))
9289, 91oveq12d 6830 . . . . . . . . . . . . . 14 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎)) = (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))
9385, 92mpteq12dv 4880 . . . . . . . . . . . . 13 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑧 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎))) = (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
9482, 93syl5eq 2820 . . . . . . . . . . . 12 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑥 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))) = (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
95 nfcv 2916 . . . . . . . . . . . . 13 𝑎(𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))
96 nfcv 2916 . . . . . . . . . . . . 13 𝑏(𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))
97 nfcv 2916 . . . . . . . . . . . . 13 𝑛(𝑥 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)))
98 nfcv 2916 . . . . . . . . . . . . 13 𝑓(𝑥 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)))
99 suceq 5944 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → suc 𝑛 = suc 𝑎)
10099adantr 467 . . . . . . . . . . . . . . 15 ((𝑛 = 𝑎𝑓 = 𝑏) → suc 𝑛 = suc 𝑎)
101100oveq2d 6828 . . . . . . . . . . . . . 14 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝐴𝑚 suc 𝑛) = (𝐴𝑚 suc 𝑎))
102 simpr 472 . . . . . . . . . . . . . . . 16 ((𝑛 = 𝑎𝑓 = 𝑏) → 𝑓 = 𝑏)
103 reseq2 5541 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → (𝑥𝑛) = (𝑥𝑎))
104103adantr 467 . . . . . . . . . . . . . . . 16 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑥𝑛) = (𝑥𝑎))
105102, 104fveq12d 6355 . . . . . . . . . . . . . . 15 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑓‘(𝑥𝑛)) = (𝑏‘(𝑥𝑎)))
106 simpl 469 . . . . . . . . . . . . . . . 16 ((𝑛 = 𝑎𝑓 = 𝑏) → 𝑛 = 𝑎)
107106fveq2d 6352 . . . . . . . . . . . . . . 15 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑥𝑛) = (𝑥𝑎))
108105, 107oveq12d 6830 . . . . . . . . . . . . . 14 ((𝑛 = 𝑎𝑓 = 𝑏) → ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)) = ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)))
109101, 108mpteq12dv 4880 . . . . . . . . . . . . 13 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))) = (𝑥 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))))
11095, 96, 97, 98, 109cbvmpt2 6902 . . . . . . . . . . . 12 (𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))))
111 ovex 6844 . . . . . . . . . . . . 13 (𝐴𝑚 suc 𝑚) ∈ V
112111mptex 6649 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))) ∈ V
11394, 110, 112ovmpt2a 6959 . . . . . . . . . . 11 ((𝑚 ∈ V ∧ (𝐺𝑚) ∈ V) → (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))))(𝐺𝑚)) = (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
11465, 77, 113mp2an 673 . . . . . . . . . 10 (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))))(𝐺𝑚)) = (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))
11576, 114syl6eq 2824 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → (𝐺‘suc 𝑚) = (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
116115feq1d 6181 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → ((𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)⟶𝐴 ↔ (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))):(𝐴𝑚 suc 𝑚)⟶𝐴))
11774, 116mpbird 248 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → (𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)⟶𝐴)
118 elmapi 8052 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝐴𝑚 suc 𝑚) → 𝑎:suc 𝑚𝐴)
119118ad2antrl 708 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝑎:suc 𝑚𝐴)
120119ffnd 6197 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝑎 Fn suc 𝑚)
121 elmapi 8052 . . . . . . . . . . . . . 14 (𝑏 ∈ (𝐴𝑚 suc 𝑚) → 𝑏:suc 𝑚𝐴)
122121ad2antll 709 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝑏:suc 𝑚𝐴)
123122ffnd 6197 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝑏 Fn suc 𝑚)
12461a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝑚 ⊆ suc 𝑚)
125 fvreseq 6479 . . . . . . . . . . . 12 (((𝑎 Fn suc 𝑚𝑏 Fn suc 𝑚) ∧ 𝑚 ⊆ suc 𝑚) → ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥)))
126120, 123, 124, 125syl21anc 855 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥)))
127 fveq2 6348 . . . . . . . . . . . . . . 15 (𝑥 = 𝑚 → (𝑎𝑥) = (𝑎𝑚))
128 fveq2 6348 . . . . . . . . . . . . . . 15 (𝑥 = 𝑚 → (𝑏𝑥) = (𝑏𝑚))
129127, 128eqeq12d 2789 . . . . . . . . . . . . . 14 (𝑥 = 𝑚 → ((𝑎𝑥) = (𝑏𝑥) ↔ (𝑎𝑚) = (𝑏𝑚)))
13065, 129ralsn 4371 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥) ↔ (𝑎𝑚) = (𝑏𝑚))
131130bicomi 215 . . . . . . . . . . . 12 ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥))
132131a1i 11 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥)))
133126, 132anbi12d 617 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (((𝑎𝑚) = (𝑏𝑚) ∧ (𝑎𝑚) = (𝑏𝑚)) ↔ (∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥))))
134115adantr 467 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝐺‘suc 𝑚) = (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
135134fveq1d 6350 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑎) = ((𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑎))
136 reseq1 5540 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑎 → (𝑧𝑚) = (𝑎𝑚))
137136fveq2d 6352 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑎 → ((𝐺𝑚)‘(𝑧𝑚)) = ((𝐺𝑚)‘(𝑎𝑚)))
138 fveq1 6347 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑎 → (𝑧𝑚) = (𝑎𝑚))
139137, 138oveq12d 6830 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑎 → (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)) = (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)))
140 eqid 2774 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))) = (𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))
141 ovex 6844 . . . . . . . . . . . . . . . 16 (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)) ∈ V
142139, 140, 141fvmpt 6441 . . . . . . . . . . . . . . 15 (𝑎 ∈ (𝐴𝑚 suc 𝑚) → ((𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑎) = (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)))
143142ad2antrl 708 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑎) = (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)))
144135, 143eqtrd 2808 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑎) = (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)))
145 df-ov 6815 . . . . . . . . . . . . 13 (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)) = (𝐹‘⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩)
146144, 145syl6eq 2824 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑎) = (𝐹‘⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩))
147134fveq1d 6350 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑏) = ((𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑏))
148 reseq1 5540 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑏 → (𝑧𝑚) = (𝑏𝑚))
149148fveq2d 6352 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑏 → ((𝐺𝑚)‘(𝑧𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)))
150 fveq1 6347 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑏 → (𝑧𝑚) = (𝑏𝑚))
151149, 150oveq12d 6830 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑏 → (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)) = (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)))
152 ovex 6844 . . . . . . . . . . . . . . . 16 (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)) ∈ V
153151, 140, 152fvmpt 6441 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝐴𝑚 suc 𝑚) → ((𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑏) = (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)))
154153ad2antll 709 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝑧 ∈ (𝐴𝑚 suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑏) = (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)))
155147, 154eqtrd 2808 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑏) = (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)))
156 df-ov 6815 . . . . . . . . . . . . 13 (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)) = (𝐹‘⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩)
157155, 156syl6eq 2824 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑏) = (𝐹‘⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩))
158146, 157eqeq12d 2789 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) ↔ (𝐹‘⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩) = (𝐹‘⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩)))
15952ad2antrr 706 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
160 f1of1 6292 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐴)–1-1-onto𝐴𝐹:(𝐴 × 𝐴)–1-1𝐴)
161159, 160syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝐹:(𝐴 × 𝐴)–1-1𝐴)
16257adantr 467 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝐺𝑚):(𝐴𝑚 𝑚)⟶𝐴)
163 fssres 6225 . . . . . . . . . . . . . . . . 17 ((𝑎:suc 𝑚𝐴𝑚 ⊆ suc 𝑚) → (𝑎𝑚):𝑚𝐴)
164119, 61, 163sylancl 575 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑎𝑚):𝑚𝐴)
16544ad2antrr 706 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → 𝐴𝑉)
166 elmapg 8043 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝑚 ∈ V) → ((𝑎𝑚) ∈ (𝐴𝑚 𝑚) ↔ (𝑎𝑚):𝑚𝐴))
167165, 65, 166sylancl 575 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝑎𝑚) ∈ (𝐴𝑚 𝑚) ↔ (𝑎𝑚):𝑚𝐴))
168164, 167mpbird 248 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑎𝑚) ∈ (𝐴𝑚 𝑚))
169162, 168ffvelrnd 6520 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺𝑚)‘(𝑎𝑚)) ∈ 𝐴)
170 ffvelrn 6517 . . . . . . . . . . . . . . 15 ((𝑎:suc 𝑚𝐴𝑚 ∈ suc 𝑚) → (𝑎𝑚) ∈ 𝐴)
171119, 70, 170sylancl 575 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑎𝑚) ∈ 𝐴)
172 opelxpi 5300 . . . . . . . . . . . . . 14 ((((𝐺𝑚)‘(𝑎𝑚)) ∈ 𝐴 ∧ (𝑎𝑚) ∈ 𝐴) → ⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ ∈ (𝐴 × 𝐴))
173169, 171, 172syl2anc 574 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ ∈ (𝐴 × 𝐴))
174 fssres 6225 . . . . . . . . . . . . . . . . 17 ((𝑏:suc 𝑚𝐴𝑚 ⊆ suc 𝑚) → (𝑏𝑚):𝑚𝐴)
175122, 61, 174sylancl 575 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑏𝑚):𝑚𝐴)
176 elmapg 8043 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝑚 ∈ V) → ((𝑏𝑚) ∈ (𝐴𝑚 𝑚) ↔ (𝑏𝑚):𝑚𝐴))
177165, 65, 176sylancl 575 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝑏𝑚) ∈ (𝐴𝑚 𝑚) ↔ (𝑏𝑚):𝑚𝐴))
178175, 177mpbird 248 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑏𝑚) ∈ (𝐴𝑚 𝑚))
179162, 178ffvelrnd 6520 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐺𝑚)‘(𝑏𝑚)) ∈ 𝐴)
180 ffvelrn 6517 . . . . . . . . . . . . . . 15 ((𝑏:suc 𝑚𝐴𝑚 ∈ suc 𝑚) → (𝑏𝑚) ∈ 𝐴)
181122, 70, 180sylancl 575 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑏𝑚) ∈ 𝐴)
182 opelxpi 5300 . . . . . . . . . . . . . 14 ((((𝐺𝑚)‘(𝑏𝑚)) ∈ 𝐴 ∧ (𝑏𝑚) ∈ 𝐴) → ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩ ∈ (𝐴 × 𝐴))
183179, 181, 182syl2anc 574 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩ ∈ (𝐴 × 𝐴))
184 f1fveq 6681 . . . . . . . . . . . . 13 ((𝐹:(𝐴 × 𝐴)–1-1𝐴 ∧ (⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ ∈ (𝐴 × 𝐴) ∧ ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩ ∈ (𝐴 × 𝐴))) → ((𝐹‘⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩) = (𝐹‘⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩) ↔ ⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ = ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩))
185161, 173, 183, 184syl12anc 854 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐹‘⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩) = (𝐹‘⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩) ↔ ⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ = ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩))
186 fvex 6359 . . . . . . . . . . . . 13 ((𝐺𝑚)‘(𝑎𝑚)) ∈ V
187 fvex 6359 . . . . . . . . . . . . 13 (𝑎𝑚) ∈ V
188186, 187opth 5086 . . . . . . . . . . . 12 (⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ = ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩ ↔ (((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ∧ (𝑎𝑚) = (𝑏𝑚)))
189185, 188syl6bb 277 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((𝐹‘⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩) = (𝐹‘⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩) ↔ (((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ∧ (𝑎𝑚) = (𝑏𝑚))))
190 simplrr 785 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)
191 f1fveq 6681 . . . . . . . . . . . . 13 (((𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴 ∧ ((𝑎𝑚) ∈ (𝐴𝑚 𝑚) ∧ (𝑏𝑚) ∈ (𝐴𝑚 𝑚))) → (((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ↔ (𝑎𝑚) = (𝑏𝑚)))
192190, 168, 178, 191syl12anc 854 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ↔ (𝑎𝑚) = (𝑏𝑚)))
193192anbi1d 616 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → ((((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ∧ (𝑎𝑚) = (𝑏𝑚)) ↔ ((𝑎𝑚) = (𝑏𝑚) ∧ (𝑎𝑚) = (𝑏𝑚))))
194158, 189, 1933bitrd 295 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) ↔ ((𝑎𝑚) = (𝑏𝑚) ∧ (𝑎𝑚) = (𝑏𝑚))))
195 eqfnfv 6471 . . . . . . . . . . . 12 ((𝑎 Fn suc 𝑚𝑏 Fn suc 𝑚) → (𝑎 = 𝑏 ↔ ∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥)))
196120, 123, 195syl2anc 574 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑎 = 𝑏 ↔ ∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥)))
197 df-suc 5883 . . . . . . . . . . . . 13 suc 𝑚 = (𝑚 ∪ {𝑚})
198197raleqi 3295 . . . . . . . . . . . 12 (∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥) ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑚})(𝑎𝑥) = (𝑏𝑥))
199 ralunb 3952 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑚 ∪ {𝑚})(𝑎𝑥) = (𝑏𝑥) ↔ (∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥)))
200198, 199bitri 265 . . . . . . . . . . 11 (∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥) ↔ (∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥)))
201196, 200syl6bb 277 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (𝑎 = 𝑏 ↔ (∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥))))
202133, 194, 2013bitr4d 301 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) ↔ 𝑎 = 𝑏))
203202biimpd 220 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴𝑚 suc 𝑚) ∧ 𝑏 ∈ (𝐴𝑚 suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) → 𝑎 = 𝑏))
204203ralrimivva 3123 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → ∀𝑎 ∈ (𝐴𝑚 suc 𝑚)∀𝑏 ∈ (𝐴𝑚 suc 𝑚)(((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) → 𝑎 = 𝑏))
205 dff13 6674 . . . . . . 7 ((𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)–1-1𝐴 ↔ ((𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)⟶𝐴 ∧ ∀𝑎 ∈ (𝐴𝑚 suc 𝑚)∀𝑏 ∈ (𝐴𝑚 suc 𝑚)(((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) → 𝑎 = 𝑏)))
206117, 204, 205sylanbrc 573 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴)) → (𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)–1-1𝐴)
207206expr 445 . . . . 5 ((𝜑𝑚 ∈ ω) → ((𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴 → (𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)–1-1𝐴))
208207expcom 399 . . . 4 (𝑚 ∈ ω → (𝜑 → ((𝐺𝑚):(𝐴𝑚 𝑚)–1-1𝐴 → (𝐺‘suc 𝑚):(𝐴𝑚 suc 𝑚)–1-1𝐴)))
20920, 27, 34, 51, 208finds2 7262 . . 3 (𝑦 ∈ ω → (𝜑 → (𝐺𝑦):(𝐴𝑚 𝑦)–1-1𝐴))
2108, 209vtoclga 3428 . 2 (𝐶 ∈ ω → (𝜑 → (𝐺𝐶):(𝐴𝑚 𝐶)–1-1𝐴))
211210impcom 395 1 ((𝜑𝐶 ∈ ω) → (𝐺𝐶):(𝐴𝑚 𝐶)–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383   = wceq 1634  wcel 2148  wral 3064  Vcvv 3355  cun 3727  wss 3729  c0 4073  {csn 4326  cop 4332  cmpt 4876   × cxp 5261  cres 5265  suc csuc 5879   Fn wfn 6037  wf 6038  1-1wf1 6039  1-1-ontowf1o 6041  cfv 6042  (class class class)co 6812  cmpt2 6814  ωcom 7233  seq𝜔cseqom 7716  1𝑜c1o 7727  𝑚 cmap 8030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-seqom 7717  df-1o 7734  df-map 8032
This theorem is referenced by:  fseqenlem2  9069
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