Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fseqenlem1 Structured version   Visualization version   GIF version

Theorem fseqenlem1 9448
 Description: Lemma for fseqen 9451. (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 6661 . . . . . 6 (𝑦 = 𝐶 → (𝐺𝑦) = (𝐺𝐶))
2 f1eq1 6560 . . . . . 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 7157 . . . . . 6 (𝑦 = 𝐶 → (𝐴m 𝑦) = (𝐴m 𝐶))
5 f1eq2 6561 . . . . . 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 282 . . . 4 (𝑦 = 𝐶 → ((𝐺𝑦):(𝐴m 𝑦)–1-1𝐴 ↔ (𝐺𝐶):(𝐴m 𝐶)–1-1𝐴))
87imbi2d 344 . . 3 (𝑦 = 𝐶 → ((𝜑 → (𝐺𝑦):(𝐴m 𝑦)–1-1𝐴) ↔ (𝜑 → (𝐺𝐶):(𝐴m 𝐶)–1-1𝐴)))
9 fveq2 6661 . . . . . . 7 (𝑦 = ∅ → (𝐺𝑦) = (𝐺‘∅))
10 snex 5319 . . . . . . . 8 {⟨∅, 𝐵⟩} ∈ V
11 fseqenlem.g . . . . . . . . 9 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
1211seqom0g 8088 . . . . . . . 8 ({⟨∅, 𝐵⟩} ∈ V → (𝐺‘∅) = {⟨∅, 𝐵⟩})
1310, 12ax-mp 5 . . . . . . 7 (𝐺‘∅) = {⟨∅, 𝐵⟩}
149, 13syl6eq 2875 . . . . . 6 (𝑦 = ∅ → (𝐺𝑦) = {⟨∅, 𝐵⟩})
15 f1eq1 6560 . . . . . 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 7157 . . . . . 6 (𝑦 = ∅ → (𝐴m 𝑦) = (𝐴m ∅))
18 f1eq2 6561 . . . . . 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 282 . . . 4 (𝑦 = ∅ → ((𝐺𝑦):(𝐴m 𝑦)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴m ∅)–1-1𝐴))
21 fveq2 6661 . . . . . 6 (𝑦 = 𝑚 → (𝐺𝑦) = (𝐺𝑚))
22 f1eq1 6560 . . . . . 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 7157 . . . . . 6 (𝑦 = 𝑚 → (𝐴m 𝑦) = (𝐴m 𝑚))
25 f1eq2 6561 . . . . . 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 282 . . . 4 (𝑦 = 𝑚 → ((𝐺𝑦):(𝐴m 𝑦)–1-1𝐴 ↔ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴))
28 fveq2 6661 . . . . . 6 (𝑦 = suc 𝑚 → (𝐺𝑦) = (𝐺‘suc 𝑚))
29 f1eq1 6560 . . . . . 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 7157 . . . . . 6 (𝑦 = suc 𝑚 → (𝐴m 𝑦) = (𝐴m suc 𝑚))
32 f1eq2 6561 . . . . . 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 282 . . . 4 (𝑦 = suc 𝑚 → ((𝐺𝑦):(𝐴m 𝑦)–1-1𝐴 ↔ (𝐺‘suc 𝑚):(𝐴m suc 𝑚)–1-1𝐴))
35 0ex 5197 . . . . . . . 8 ∅ ∈ V
36 fseqenlem.b . . . . . . . 8 (𝜑𝐵𝐴)
37 f1osng 6646 . . . . . . . 8 ((∅ ∈ V ∧ 𝐵𝐴) → {⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵})
3835, 36, 37sylancr 590 . . . . . . 7 (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵})
39 f1of1 6605 . . . . . . 7 ({⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵} → {⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵})
4038, 39syl 17 . . . . . 6 (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵})
4136snssd 4726 . . . . . 6 (𝜑 → {𝐵} ⊆ 𝐴)
42 f1ss 6571 . . . . . 6 (({⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝐴) → {⟨∅, 𝐵⟩}:{∅}–1-1𝐴)
4340, 41, 42syl2anc 587 . . . . 5 (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1𝐴)
44 fseqenlem.a . . . . . . . 8 (𝜑𝐴𝑉)
45 map0e 8442 . . . . . . . 8 (𝐴𝑉 → (𝐴m ∅) = 1o)
4644, 45syl 17 . . . . . . 7 (𝜑 → (𝐴m ∅) = 1o)
47 df1o2 8112 . . . . . . 7 1o = {∅}
4846, 47syl6eq 2875 . . . . . 6 (𝜑 → (𝐴m ∅) = {∅})
49 f1eq2 6561 . . . . . 6 ((𝐴m ∅) = {∅} → ({⟨∅, 𝐵⟩}:(𝐴m ∅)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:{∅}–1-1𝐴))
5048, 49syl 17 . . . . 5 (𝜑 → ({⟨∅, 𝐵⟩}:(𝐴m ∅)–1-1𝐴 ↔ {⟨∅, 𝐵⟩}:{∅}–1-1𝐴))
5143, 50mpbird 260 . . . 4 (𝜑 → {⟨∅, 𝐵⟩}:(𝐴m ∅)–1-1𝐴)
5211seqomsuc 8089 . . . . . . . . . 10 (𝑚 ∈ ω → (𝐺‘suc 𝑚) = (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))))(𝐺𝑚)))
5352ad2antrl 727 . . . . . . . . 9 ((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)))
54 vex 3483 . . . . . . . . . 10 𝑚 ∈ V
55 fvex 6674 . . . . . . . . . 10 (𝐺𝑚) ∈ V
56 reseq1 5834 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑥𝑎) = (𝑧𝑎))
5756fveq2d 6665 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑏‘(𝑥𝑎)) = (𝑏‘(𝑧𝑎)))
58 fveq1 6660 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑥𝑎) = (𝑧𝑎))
5957, 58oveq12d 7167 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)) = ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎)))
6059cbvmptv 5155 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))) = (𝑧 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎)))
61 suceq 6243 . . . . . . . . . . . . . . 15 (𝑎 = 𝑚 → suc 𝑎 = suc 𝑚)
6261adantr 484 . . . . . . . . . . . . . 14 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → suc 𝑎 = suc 𝑚)
6362oveq2d 7165 . . . . . . . . . . . . 13 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝐴m suc 𝑎) = (𝐴m suc 𝑚))
64 simpr 488 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → 𝑏 = (𝐺𝑚))
65 reseq2 5835 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑚 → (𝑧𝑎) = (𝑧𝑚))
6665adantr 484 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑧𝑎) = (𝑧𝑚))
6764, 66fveq12d 6668 . . . . . . . . . . . . . 14 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑏‘(𝑧𝑎)) = ((𝐺𝑚)‘(𝑧𝑚)))
68 simpl 486 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → 𝑎 = 𝑚)
6968fveq2d 6665 . . . . . . . . . . . . . 14 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑧𝑎) = (𝑧𝑚))
7067, 69oveq12d 7167 . . . . . . . . . . . . 13 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎)) = (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))
7163, 70mpteq12dv 5137 . . . . . . . . . . . 12 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑧 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑧𝑎))𝐹(𝑧𝑎))) = (𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
7260, 71syl5eq 2871 . . . . . . . . . . 11 ((𝑎 = 𝑚𝑏 = (𝐺𝑚)) → (𝑥 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))) = (𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
73 nfcv 2982 . . . . . . . . . . . 12 𝑎(𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))
74 nfcv 2982 . . . . . . . . . . . 12 𝑏(𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))
75 nfcv 2982 . . . . . . . . . . . 12 𝑛(𝑥 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)))
76 nfcv 2982 . . . . . . . . . . . 12 𝑓(𝑥 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)))
77 suceq 6243 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → suc 𝑛 = suc 𝑎)
7877adantr 484 . . . . . . . . . . . . . 14 ((𝑛 = 𝑎𝑓 = 𝑏) → suc 𝑛 = suc 𝑎)
7978oveq2d 7165 . . . . . . . . . . . . 13 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝐴m suc 𝑛) = (𝐴m suc 𝑎))
80 simpr 488 . . . . . . . . . . . . . . 15 ((𝑛 = 𝑎𝑓 = 𝑏) → 𝑓 = 𝑏)
81 reseq2 5835 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → (𝑥𝑛) = (𝑥𝑎))
8281adantr 484 . . . . . . . . . . . . . . 15 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑥𝑛) = (𝑥𝑎))
8380, 82fveq12d 6668 . . . . . . . . . . . . . 14 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑓‘(𝑥𝑛)) = (𝑏‘(𝑥𝑎)))
84 simpl 486 . . . . . . . . . . . . . . 15 ((𝑛 = 𝑎𝑓 = 𝑏) → 𝑛 = 𝑎)
8584fveq2d 6665 . . . . . . . . . . . . . 14 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑥𝑛) = (𝑥𝑎))
8683, 85oveq12d 7167 . . . . . . . . . . . . 13 ((𝑛 = 𝑎𝑓 = 𝑏) → ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)) = ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎)))
8779, 86mpteq12dv 5137 . . . . . . . . . . . 12 ((𝑛 = 𝑎𝑓 = 𝑏) → (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))) = (𝑥 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))))
8873, 74, 75, 76, 87cbvmpo 7241 . . . . . . . . . . 11 (𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑎) ↦ ((𝑏‘(𝑥𝑎))𝐹(𝑥𝑎))))
89 ovex 7182 . . . . . . . . . . . 12 (𝐴m suc 𝑚) ∈ V
9089mptex 6977 . . . . . . . . . . 11 (𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))) ∈ V
9172, 88, 90ovmpoa 7298 . . . . . . . . . 10 ((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))))
9254, 55, 91mp2an 691 . . . . . . . . 9 (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛))))(𝐺𝑚)) = (𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))
9353, 92syl6eq 2875 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) → (𝐺‘suc 𝑚) = (𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))))
94 fseqenlem.f . . . . . . . . . . 11 (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
9594ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → 𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
96 f1of 6606 . . . . . . . . . 10 (𝐹:(𝐴 × 𝐴)–1-1-onto𝐴𝐹:(𝐴 × 𝐴)⟶𝐴)
9795, 96syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → 𝐹:(𝐴 × 𝐴)⟶𝐴)
98 f1f 6565 . . . . . . . . . . . 12 ((𝐺𝑚):(𝐴m 𝑚)–1-1𝐴 → (𝐺𝑚):(𝐴m 𝑚)⟶𝐴)
9998ad2antll 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) → (𝐺𝑚):(𝐴m 𝑚)⟶𝐴)
10099adantr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → (𝐺𝑚):(𝐴m 𝑚)⟶𝐴)
101 elmapi 8424 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐴m suc 𝑚) → 𝑧:suc 𝑚𝐴)
102101adantl 485 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → 𝑧:suc 𝑚𝐴)
103 sssucid 6255 . . . . . . . . . . . 12 𝑚 ⊆ suc 𝑚
104 fssres 6534 . . . . . . . . . . . 12 ((𝑧:suc 𝑚𝐴𝑚 ⊆ suc 𝑚) → (𝑧𝑚):𝑚𝐴)
105102, 103, 104sylancl 589 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → (𝑧𝑚):𝑚𝐴)
10644ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → 𝐴𝑉)
107 elmapg 8415 . . . . . . . . . . . 12 ((𝐴𝑉𝑚 ∈ V) → ((𝑧𝑚) ∈ (𝐴m 𝑚) ↔ (𝑧𝑚):𝑚𝐴))
108106, 54, 107sylancl 589 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → ((𝑧𝑚) ∈ (𝐴m 𝑚) ↔ (𝑧𝑚):𝑚𝐴))
109105, 108mpbird 260 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → (𝑧𝑚) ∈ (𝐴m 𝑚))
110100, 109ffvelrnd 6843 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → ((𝐺𝑚)‘(𝑧𝑚)) ∈ 𝐴)
11154sucid 6257 . . . . . . . . . 10 𝑚 ∈ suc 𝑚
112 ffvelrn 6840 . . . . . . . . . 10 ((𝑧:suc 𝑚𝐴𝑚 ∈ suc 𝑚) → (𝑧𝑚) ∈ 𝐴)
113102, 111, 112sylancl 589 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → (𝑧𝑚) ∈ 𝐴)
11497, 110, 113fovrnd 7314 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ 𝑧 ∈ (𝐴m suc 𝑚)) → (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)) ∈ 𝐴)
11593, 114fmpt3d 6871 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) → (𝐺‘suc 𝑚):(𝐴m suc 𝑚)⟶𝐴)
116 elmapi 8424 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝐴m suc 𝑚) → 𝑎:suc 𝑚𝐴)
117116ad2antrl 727 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝑎:suc 𝑚𝐴)
118117ffnd 6504 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝑎 Fn suc 𝑚)
119 elmapi 8424 . . . . . . . . . . . . . 14 (𝑏 ∈ (𝐴m suc 𝑚) → 𝑏:suc 𝑚𝐴)
120119ad2antll 728 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝑏:suc 𝑚𝐴)
121120ffnd 6504 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝑏 Fn suc 𝑚)
122103a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝑚 ⊆ suc 𝑚)
123 fvreseq 6801 . . . . . . . . . . . 12 (((𝑎 Fn suc 𝑚𝑏 Fn suc 𝑚) ∧ 𝑚 ⊆ suc 𝑚) → ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥)))
124118, 121, 122, 123syl21anc 836 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥)))
125 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑥 = 𝑚 → (𝑎𝑥) = (𝑎𝑚))
126 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑥 = 𝑚 → (𝑏𝑥) = (𝑏𝑚))
127125, 126eqeq12d 2840 . . . . . . . . . . . . . 14 (𝑥 = 𝑚 → ((𝑎𝑥) = (𝑏𝑥) ↔ (𝑎𝑚) = (𝑏𝑚)))
12854, 127ralsn 4604 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥) ↔ (𝑎𝑚) = (𝑏𝑚))
129128bicomi 227 . . . . . . . . . . . 12 ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥))
130129a1i 11 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ((𝑎𝑚) = (𝑏𝑚) ↔ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥)))
131124, 130anbi12d 633 . . . . . . . . . 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))))
13293adantr 484 . . . . . . . . . . . . . . 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))))
133132fveq1d 6663 . . . . . . . . . . . . . 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))
134 reseq1 5834 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑎 → (𝑧𝑚) = (𝑎𝑚))
135134fveq2d 6665 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑎 → ((𝐺𝑚)‘(𝑧𝑚)) = ((𝐺𝑚)‘(𝑎𝑚)))
136 fveq1 6660 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑎 → (𝑧𝑚) = (𝑎𝑚))
137135, 136oveq12d 7167 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑎 → (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)) = (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)))
138 eqid 2824 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚))) = (𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))
139 ovex 7182 . . . . . . . . . . . . . . . 16 (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)) ∈ V
140137, 138, 139fvmpt 6759 . . . . . . . . . . . . . . 15 (𝑎 ∈ (𝐴m suc 𝑚) → ((𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑎) = (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)))
141140ad2antrl 727 . . . . . . . . . . . . . 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)))
142133, 141eqtrd 2859 . . . . . . . . . . . . 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)))
143 df-ov 7152 . . . . . . . . . . . . 13 (((𝐺𝑚)‘(𝑎𝑚))𝐹(𝑎𝑚)) = (𝐹‘⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩)
144142, 143syl6eq 2875 . . . . . . . . . . . 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)⟩))
145132fveq1d 6663 . . . . . . . . . . . . . 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))
146 reseq1 5834 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑏 → (𝑧𝑚) = (𝑏𝑚))
147146fveq2d 6665 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑏 → ((𝐺𝑚)‘(𝑧𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)))
148 fveq1 6660 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑏 → (𝑧𝑚) = (𝑏𝑚))
149147, 148oveq12d 7167 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑏 → (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)) = (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)))
150 ovex 7182 . . . . . . . . . . . . . . . 16 (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)) ∈ V
151149, 138, 150fvmpt 6759 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝐴m suc 𝑚) → ((𝑧 ∈ (𝐴m suc 𝑚) ↦ (((𝐺𝑚)‘(𝑧𝑚))𝐹(𝑧𝑚)))‘𝑏) = (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)))
152151ad2antll 728 . . . . . . . . . . . . . 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)))
153145, 152eqtrd 2859 . . . . . . . . . . . . 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)))
154 df-ov 7152 . . . . . . . . . . . . 13 (((𝐺𝑚)‘(𝑏𝑚))𝐹(𝑏𝑚)) = (𝐹‘⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩)
155153, 154syl6eq 2875 . . . . . . . . . . . 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)⟩))
156144, 155eqeq12d 2840 . . . . . . . . . . 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)⟩)))
15794ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
158 f1of1 6605 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐴)–1-1-onto𝐴𝐹:(𝐴 × 𝐴)–1-1𝐴)
159157, 158syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝐹:(𝐴 × 𝐴)–1-1𝐴)
16099adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝐺𝑚):(𝐴m 𝑚)⟶𝐴)
161 fssres 6534 . . . . . . . . . . . . . . . . 17 ((𝑎:suc 𝑚𝐴𝑚 ⊆ suc 𝑚) → (𝑎𝑚):𝑚𝐴)
162117, 103, 161sylancl 589 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝑎𝑚):𝑚𝐴)
16344ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → 𝐴𝑉)
164 elmapg 8415 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝑚 ∈ V) → ((𝑎𝑚) ∈ (𝐴m 𝑚) ↔ (𝑎𝑚):𝑚𝐴))
165163, 54, 164sylancl 589 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ((𝑎𝑚) ∈ (𝐴m 𝑚) ↔ (𝑎𝑚):𝑚𝐴))
166162, 165mpbird 260 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝑎𝑚) ∈ (𝐴m 𝑚))
167160, 166ffvelrnd 6843 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ((𝐺𝑚)‘(𝑎𝑚)) ∈ 𝐴)
168 ffvelrn 6840 . . . . . . . . . . . . . . 15 ((𝑎:suc 𝑚𝐴𝑚 ∈ suc 𝑚) → (𝑎𝑚) ∈ 𝐴)
169117, 111, 168sylancl 589 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝑎𝑚) ∈ 𝐴)
170167, 169opelxpd 5580 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ ∈ (𝐴 × 𝐴))
171 fssres 6534 . . . . . . . . . . . . . . . . 17 ((𝑏:suc 𝑚𝐴𝑚 ⊆ suc 𝑚) → (𝑏𝑚):𝑚𝐴)
172120, 103, 171sylancl 589 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝑏𝑚):𝑚𝐴)
173 elmapg 8415 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝑚 ∈ V) → ((𝑏𝑚) ∈ (𝐴m 𝑚) ↔ (𝑏𝑚):𝑚𝐴))
174163, 54, 173sylancl 589 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ((𝑏𝑚) ∈ (𝐴m 𝑚) ↔ (𝑏𝑚):𝑚𝐴))
175172, 174mpbird 260 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝑏𝑚) ∈ (𝐴m 𝑚))
176160, 175ffvelrnd 6843 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ((𝐺𝑚)‘(𝑏𝑚)) ∈ 𝐴)
177 ffvelrn 6840 . . . . . . . . . . . . . . 15 ((𝑏:suc 𝑚𝐴𝑚 ∈ suc 𝑚) → (𝑏𝑚) ∈ 𝐴)
178120, 111, 177sylancl 589 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝑏𝑚) ∈ 𝐴)
179176, 178opelxpd 5580 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩ ∈ (𝐴 × 𝐴))
180 f1fveq 7012 . . . . . . . . . . . . 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)⟩))
181159, 170, 179, 180syl12anc 835 . . . . . . . . . . . 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)⟩))
182 fvex 6674 . . . . . . . . . . . . 13 ((𝐺𝑚)‘(𝑎𝑚)) ∈ V
183 fvex 6674 . . . . . . . . . . . . 13 (𝑎𝑚) ∈ V
184182, 183opth 5355 . . . . . . . . . . . 12 (⟨((𝐺𝑚)‘(𝑎𝑚)), (𝑎𝑚)⟩ = ⟨((𝐺𝑚)‘(𝑏𝑚)), (𝑏𝑚)⟩ ↔ (((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ∧ (𝑎𝑚) = (𝑏𝑚)))
185181, 184syl6bb 290 . . . . . . . . . . 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))))
186 simplrr 777 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)
187 f1fveq 7012 . . . . . . . . . . . . 13 (((𝐺𝑚):(𝐴m 𝑚)–1-1𝐴 ∧ ((𝑎𝑚) ∈ (𝐴m 𝑚) ∧ (𝑏𝑚) ∈ (𝐴m 𝑚))) → (((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ↔ (𝑎𝑚) = (𝑏𝑚)))
188186, 166, 175, 187syl12anc 835 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (((𝐺𝑚)‘(𝑎𝑚)) = ((𝐺𝑚)‘(𝑏𝑚)) ↔ (𝑎𝑚) = (𝑏𝑚)))
189188anbi1d 632 . . . . . . . . . . 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))))
190156, 185, 1893bitrd 308 . . . . . . . . . 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))))
191 eqfnfv 6793 . . . . . . . . . . . 12 ((𝑎 Fn suc 𝑚𝑏 Fn suc 𝑚) → (𝑎 = 𝑏 ↔ ∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥)))
192118, 121, 191syl2anc 587 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) ∧ (𝑎 ∈ (𝐴m suc 𝑚) ∧ 𝑏 ∈ (𝐴m suc 𝑚))) → (𝑎 = 𝑏 ↔ ∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥)))
193 df-suc 6184 . . . . . . . . . . . . 13 suc 𝑚 = (𝑚 ∪ {𝑚})
194193raleqi 3400 . . . . . . . . . . . 12 (∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥) ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑚})(𝑎𝑥) = (𝑏𝑥))
195 ralunb 4153 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑚 ∪ {𝑚})(𝑎𝑥) = (𝑏𝑥) ↔ (∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥)))
196194, 195bitri 278 . . . . . . . . . . 11 (∀𝑥 ∈ suc 𝑚(𝑎𝑥) = (𝑏𝑥) ↔ (∀𝑥𝑚 (𝑎𝑥) = (𝑏𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎𝑥) = (𝑏𝑥)))
197192, 196syl6bb 290 . . . . . . . . . 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))))
198131, 190, 1973bitr4d 314 . . . . . . . . 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))
199198biimpd 232 . . . . . . . 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))
200199ralrimivva 3186 . . . . . . 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))
201 dff13 7005 . . . . . . 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)))
202115, 200, 201sylanbrc 586 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺𝑚):(𝐴m 𝑚)–1-1𝐴)) → (𝐺‘suc 𝑚):(𝐴m suc 𝑚)–1-1𝐴)
203202expr 460 . . . . 5 ((𝜑𝑚 ∈ ω) → ((𝐺𝑚):(𝐴m 𝑚)–1-1𝐴 → (𝐺‘suc 𝑚):(𝐴m suc 𝑚)–1-1𝐴))
204203expcom 417 . . . 4 (𝑚 ∈ ω → (𝜑 → ((𝐺𝑚):(𝐴m 𝑚)–1-1𝐴 → (𝐺‘suc 𝑚):(𝐴m suc 𝑚)–1-1𝐴)))
20520, 27, 34, 51, 204finds2 7605 . . 3 (𝑦 ∈ ω → (𝜑 → (𝐺𝑦):(𝐴m 𝑦)–1-1𝐴))
2068, 205vtoclga 3560 . 2 (𝐶 ∈ ω → (𝜑 → (𝐺𝐶):(𝐴m 𝐶)–1-1𝐴))
207206impcom 411 1 ((𝜑𝐶 ∈ ω) → (𝐺𝐶):(𝐴m 𝐶)–1-1𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ∪ cun 3917   ⊆ wss 3919  ∅c0 4276  {csn 4550  ⟨cop 4556   ↦ cmpt 5132   × cxp 5540   ↾ cres 5544  suc csuc 6180   Fn wfn 6338  ⟶wf 6339  –1-1→wf1 6340  –1-1-onto→wf1o 6342  ‘cfv 6343  (class class class)co 7149   ∈ cmpo 7151  ωcom 7574  seqωcseqom 8079  1oc1o 8091   ↑m cmap 8402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-seqom 8080  df-1o 8098  df-map 8404 This theorem is referenced by:  fseqenlem2  9449
 Copyright terms: Public domain W3C validator