fseqenlem1 - Metamath Proof Explorer

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

Theorem fseqenlem1 10030

Description: Lemma for fseqen 10033. (Contributed by Mario Carneiro, 17-May-2015.)

**Hypotheses**
Ref	Expression
fseqenlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fseqenlem.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
fseqenlem.f	⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
fseqenlem.g	⊢ 𝐺 = seq_ω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})

**Assertion**
Ref	Expression
fseqenlem1	⊢ ((𝜑 ∧ 𝐶 ∈ ω) → (𝐺‘𝐶):(𝐴 ↑_m 𝐶)–1-1→𝐴)

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.

Step	Hyp	Ref	Expression
1		fveq2 6872	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐺‘𝑦) = (𝐺‘𝐶))
2		f1eq1 6765	. . . . . 6 ⊢ ((𝐺‘𝑦) = (𝐺‘𝐶) → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝐶):(𝐴 ↑_m 𝑦)–1-1→𝐴))
3	1, 2	syl 17	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝐶):(𝐴 ↑_m 𝑦)–1-1→𝐴))
4		oveq2 7407	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐴 ↑_m 𝑦) = (𝐴 ↑_m 𝐶))
5		f1eq2 6766	. . . . . 6 ⊢ ((𝐴 ↑_m 𝑦) = (𝐴 ↑_m 𝐶) → ((𝐺‘𝐶):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝐶):(𝐴 ↑_m 𝐶)–1-1→𝐴))
6	4, 5	syl 17	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝐺‘𝐶):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝐶):(𝐴 ↑_m 𝐶)–1-1→𝐴))
7	3, 6	bitrd 279	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝐶):(𝐴 ↑_m 𝐶)–1-1→𝐴))
8	7	imbi2d 340	. . 3 ⊢ (𝑦 = 𝐶 → ((𝜑 → (𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴) ↔ (𝜑 → (𝐺‘𝐶):(𝐴 ↑_m 𝐶)–1-1→𝐴)))
9		fveq2 6872	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐺‘𝑦) = (𝐺‘∅))
10		snex 5403	. . . . . . . 8 ⊢ {⟨∅, 𝐵⟩} ∈ V
11		fseqenlem.g	. . . . . . . . 9 ⊢ 𝐺 = seq_ω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})
12	11	seqom0g 8464	. . . . . . . 8 ⊢ ({⟨∅, 𝐵⟩} ∈ V → (𝐺‘∅) = {⟨∅, 𝐵⟩})
13	10, 12	ax-mp 5	. . . . . . 7 ⊢ (𝐺‘∅) = {⟨∅, 𝐵⟩}
14	9, 13	eqtrdi 2785	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐺‘𝑦) = {⟨∅, 𝐵⟩})
15		f1eq1 6765	. . . . . 6 ⊢ ((𝐺‘𝑦) = {⟨∅, 𝐵⟩} → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴 ↑_m 𝑦)–1-1→𝐴))
16	14, 15	syl 17	. . . . 5 ⊢ (𝑦 = ∅ → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴 ↑_m 𝑦)–1-1→𝐴))
17		oveq2 7407	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐴 ↑_m 𝑦) = (𝐴 ↑_m ∅))
18		f1eq2 6766	. . . . . 6 ⊢ ((𝐴 ↑_m 𝑦) = (𝐴 ↑_m ∅) → ({⟨∅, 𝐵⟩}:(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴 ↑_m ∅)–1-1→𝐴))
19	17, 18	syl 17	. . . . 5 ⊢ (𝑦 = ∅ → ({⟨∅, 𝐵⟩}:(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴 ↑_m ∅)–1-1→𝐴))
20	16, 19	bitrd 279	. . . 4 ⊢ (𝑦 = ∅ → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ {⟨∅, 𝐵⟩}:(𝐴 ↑_m ∅)–1-1→𝐴))
21		fveq2 6872	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝐺‘𝑦) = (𝐺‘𝑚))
22		f1eq1 6765	. . . . . 6 ⊢ ((𝐺‘𝑦) = (𝐺‘𝑚) → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴))
23	21, 22	syl 17	. . . . 5 ⊢ (𝑦 = 𝑚 → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴))
24		oveq2 7407	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝐴 ↑_m 𝑦) = (𝐴 ↑_m 𝑚))
25		f1eq2 6766	. . . . . 6 ⊢ ((𝐴 ↑_m 𝑦) = (𝐴 ↑_m 𝑚) → ((𝐺‘𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴))
26	24, 25	syl 17	. . . . 5 ⊢ (𝑦 = 𝑚 → ((𝐺‘𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴))
27	23, 26	bitrd 279	. . . 4 ⊢ (𝑦 = 𝑚 → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴))
28		fveq2 6872	. . . . . 6 ⊢ (𝑦 = suc 𝑚 → (𝐺‘𝑦) = (𝐺‘suc 𝑚))
29		f1eq1 6765	. . . . . 6 ⊢ ((𝐺‘𝑦) = (𝐺‘suc 𝑚) → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘suc 𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴))
30	28, 29	syl 17	. . . . 5 ⊢ (𝑦 = suc 𝑚 → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘suc 𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴))
31		oveq2 7407	. . . . . 6 ⊢ (𝑦 = suc 𝑚 → (𝐴 ↑_m 𝑦) = (𝐴 ↑_m suc 𝑚))
32		f1eq2 6766	. . . . . 6 ⊢ ((𝐴 ↑_m 𝑦) = (𝐴 ↑_m suc 𝑚) → ((𝐺‘suc 𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)–1-1→𝐴))
33	31, 32	syl 17	. . . . 5 ⊢ (𝑦 = suc 𝑚 → ((𝐺‘suc 𝑚):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)–1-1→𝐴))
34	30, 33	bitrd 279	. . . 4 ⊢ (𝑦 = suc 𝑚 → ((𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴 ↔ (𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)–1-1→𝐴))
35		0ex 5274	. . . . . . . 8 ⊢ ∅ ∈ V
36		fseqenlem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
37		f1osng 6855	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝐴) → {⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵})
38	35, 36, 37	sylancr 587	. . . . . . 7 ⊢ (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵})
39		f1of1 6813	. . . . . . 7 ⊢ ({⟨∅, 𝐵⟩}:{∅}–1-1-onto→{𝐵} → {⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵})
40	38, 39	syl 17	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵})
41	36	snssd 4782	. . . . . 6 ⊢ (𝜑 → {𝐵} ⊆ 𝐴)
42		f1ss 6775	. . . . . 6 ⊢ (({⟨∅, 𝐵⟩}:{∅}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝐴) → {⟨∅, 𝐵⟩}:{∅}–1-1→𝐴)
43	40, 41, 42	syl2anc 584	. . . . 5 ⊢ (𝜑 → {⟨∅, 𝐵⟩}:{∅}–1-1→𝐴)
44		fseqenlem.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
45		map0e 8890	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑_m ∅) = 1_o)
46	44, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_m ∅) = 1_o)
47		df1o2 8481	. . . . . . 7 ⊢ 1_o = {∅}
48	46, 47	eqtrdi 2785	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑_m ∅) = {∅})
49		f1eq2 6766	. . . . . 6 ⊢ ((𝐴 ↑_m ∅) = {∅} → ({⟨∅, 𝐵⟩}:(𝐴 ↑_m ∅)–1-1→𝐴 ↔ {⟨∅, 𝐵⟩}:{∅}–1-1→𝐴))
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → ({⟨∅, 𝐵⟩}:(𝐴 ↑_m ∅)–1-1→𝐴 ↔ {⟨∅, 𝐵⟩}:{∅}–1-1→𝐴))
51	43, 50	mpbird 257	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐵⟩}:(𝐴 ↑_m ∅)–1-1→𝐴)
52	11	seqomsuc 8465	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → (𝐺‘suc 𝑚) = (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛))))(𝐺‘𝑚)))
53	52	ad2antrl 728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) → (𝐺‘suc 𝑚) = (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛))))(𝐺‘𝑚)))
54		vex 3461	. . . . . . . . . 10 ⊢ 𝑚 ∈ V
55		fvex 6885	. . . . . . . . . 10 ⊢ (𝐺‘𝑚) ∈ V
56		reseq1 5957	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑥 ↾ 𝑎) = (𝑧 ↾ 𝑎))
57	56	fveq2d 6876	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑏‘(𝑥 ↾ 𝑎)) = (𝑏‘(𝑧 ↾ 𝑎)))
58		fveq1 6871	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑎) = (𝑧‘𝑎))
59	57, 58	oveq12d 7417	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎)) = ((𝑏‘(𝑧 ↾ 𝑎))𝐹(𝑧‘𝑎)))
60	59	cbvmptv 5222	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎))) = (𝑧 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑧 ↾ 𝑎))𝐹(𝑧‘𝑎)))
61		suceq 6416	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑚 → suc 𝑎 = suc 𝑚)
62	61	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → suc 𝑎 = suc 𝑚)
63	62	oveq2d 7415	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → (𝐴 ↑_m suc 𝑎) = (𝐴 ↑_m suc 𝑚))
64		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → 𝑏 = (𝐺‘𝑚))
65		reseq2 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑚 → (𝑧 ↾ 𝑎) = (𝑧 ↾ 𝑚))
66	65	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → (𝑧 ↾ 𝑎) = (𝑧 ↾ 𝑚))
67	64, 66	fveq12d 6879	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → (𝑏‘(𝑧 ↾ 𝑎)) = ((𝐺‘𝑚)‘(𝑧 ↾ 𝑚)))
68		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → 𝑎 = 𝑚)
69	68	fveq2d 6876	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → (𝑧‘𝑎) = (𝑧‘𝑚))
70	67, 69	oveq12d 7417	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → ((𝑏‘(𝑧 ↾ 𝑎))𝐹(𝑧‘𝑎)) = (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))
71	63, 70	mpteq12dv 5204	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → (𝑧 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑧 ↾ 𝑎))𝐹(𝑧‘𝑎))) = (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚))))
72	60, 71	eqtrid 2781	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑚 ∧ 𝑏 = (𝐺‘𝑚)) → (𝑥 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎))) = (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚))))
73		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))
74		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏(𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))
75		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑥 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎)))
76		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓(𝑥 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎)))
77		suceq 6416	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → suc 𝑛 = suc 𝑎)
78	77	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → suc 𝑛 = suc 𝑎)
79	78	oveq2d 7415	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → (𝐴 ↑_m suc 𝑛) = (𝐴 ↑_m suc 𝑎))
80		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → 𝑓 = 𝑏)
81		reseq2 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → (𝑥 ↾ 𝑛) = (𝑥 ↾ 𝑎))
82	81	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → (𝑥 ↾ 𝑛) = (𝑥 ↾ 𝑎))
83	80, 82	fveq12d 6879	. . . . . . . . . . . . . 14 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → (𝑓‘(𝑥 ↾ 𝑛)) = (𝑏‘(𝑥 ↾ 𝑎)))
84		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → 𝑛 = 𝑎)
85	84	fveq2d 6876	. . . . . . . . . . . . . 14 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → (𝑥‘𝑛) = (𝑥‘𝑎))
86	83, 85	oveq12d 7417	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)) = ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎)))
87	79, 86	mpteq12dv 5204	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑎 ∧ 𝑓 = 𝑏) → (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛))) = (𝑥 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎))))
88	73, 74, 75, 76, 87	cbvmpo 7495	. . . . . . . . . . 11 ⊢ (𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑎) ↦ ((𝑏‘(𝑥 ↾ 𝑎))𝐹(𝑥‘𝑎))))
89		ovex 7432	. . . . . . . . . . . 12 ⊢ (𝐴 ↑_m suc 𝑚) ∈ V
90	89	mptex 7211	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚))) ∈ V
91	72, 88, 90	ovmpoa 7556	. . . . . . . . . 10 ⊢ ((𝑚 ∈ V ∧ (𝐺‘𝑚) ∈ V) → (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛))))(𝐺‘𝑚)) = (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚))))
92	54, 55, 91	mp2an 692	. . . . . . . . 9 ⊢ (𝑚(𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛))))(𝐺‘𝑚)) = (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))
93	53, 92	eqtrdi 2785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) → (𝐺‘suc 𝑚) = (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚))))
94		fseqenlem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
95	94	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
96		f1of 6814	. . . . . . . . . 10 ⊢ (𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴 → 𝐹:(𝐴 × 𝐴)⟶𝐴)
97	95, 96	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → 𝐹:(𝐴 × 𝐴)⟶𝐴)
98		f1f 6770	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴 → (𝐺‘𝑚):(𝐴 ↑_m 𝑚)⟶𝐴)
99	98	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) → (𝐺‘𝑚):(𝐴 ↑_m 𝑚)⟶𝐴)
100	99	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → (𝐺‘𝑚):(𝐴 ↑_m 𝑚)⟶𝐴)
101		elmapi 8857	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝐴 ↑_m suc 𝑚) → 𝑧:suc 𝑚⟶𝐴)
102	101	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → 𝑧:suc 𝑚⟶𝐴)
103		sssucid 6430	. . . . . . . . . . . 12 ⊢ 𝑚 ⊆ suc 𝑚
104		fssres 6740	. . . . . . . . . . . 12 ⊢ ((𝑧:suc 𝑚⟶𝐴 ∧ 𝑚 ⊆ suc 𝑚) → (𝑧 ↾ 𝑚):𝑚⟶𝐴)
105	102, 103, 104	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → (𝑧 ↾ 𝑚):𝑚⟶𝐴)
106	44	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → 𝐴 ∈ 𝑉)
107		elmapg 8847	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑚 ∈ V) → ((𝑧 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚) ↔ (𝑧 ↾ 𝑚):𝑚⟶𝐴))
108	106, 54, 107	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → ((𝑧 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚) ↔ (𝑧 ↾ 𝑚):𝑚⟶𝐴))
109	105, 108	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → (𝑧 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚))
110	100, 109	ffvelcdmd 7071	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → ((𝐺‘𝑚)‘(𝑧 ↾ 𝑚)) ∈ 𝐴)
111	54	sucid 6432	. . . . . . . . . 10 ⊢ 𝑚 ∈ suc 𝑚
112		ffvelcdm 7067	. . . . . . . . . 10 ⊢ ((𝑧:suc 𝑚⟶𝐴 ∧ 𝑚 ∈ suc 𝑚) → (𝑧‘𝑚) ∈ 𝐴)
113	102, 111, 112	sylancl 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → (𝑧‘𝑚) ∈ 𝐴)
114	97, 110, 113	fovcdmd 7573	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ 𝑧 ∈ (𝐴 ↑_m suc 𝑚)) → (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)) ∈ 𝐴)
115	93, 114	fmpt3d 7102	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) → (𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)⟶𝐴)
116		elmapi 8857	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) → 𝑎:suc 𝑚⟶𝐴)
117	116	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝑎:suc 𝑚⟶𝐴)
118	117	ffnd 6703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝑎 Fn suc 𝑚)
119		elmapi 8857	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝐴 ↑_m suc 𝑚) → 𝑏:suc 𝑚⟶𝐴)
120	119	ad2antll 729	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝑏:suc 𝑚⟶𝐴)
121	120	ffnd 6703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝑏 Fn suc 𝑚)
122	103	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝑚 ⊆ suc 𝑚)
123		fvreseq 7026	. . . . . . . . . . . 12 ⊢ (((𝑎 Fn suc 𝑚 ∧ 𝑏 Fn suc 𝑚) ∧ 𝑚 ⊆ suc 𝑚) → ((𝑎 ↾ 𝑚) = (𝑏 ↾ 𝑚) ↔ ∀𝑥 ∈ 𝑚 (𝑎‘𝑥) = (𝑏‘𝑥)))
124	118, 121, 122, 123	syl21anc 837	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝑎 ↾ 𝑚) = (𝑏 ↾ 𝑚) ↔ ∀𝑥 ∈ 𝑚 (𝑎‘𝑥) = (𝑏‘𝑥)))
125		fveq2 6872	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑚 → (𝑎‘𝑥) = (𝑎‘𝑚))
126		fveq2 6872	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑚 → (𝑏‘𝑥) = (𝑏‘𝑚))
127	125, 126	eqeq12d 2750	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑚 → ((𝑎‘𝑥) = (𝑏‘𝑥) ↔ (𝑎‘𝑚) = (𝑏‘𝑚)))
128	54, 127	ralsn 4654	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑚} (𝑎‘𝑥) = (𝑏‘𝑥) ↔ (𝑎‘𝑚) = (𝑏‘𝑚))
129	128	bicomi 224	. . . . . . . . . . . 12 ⊢ ((𝑎‘𝑚) = (𝑏‘𝑚) ↔ ∀𝑥 ∈ {𝑚} (𝑎‘𝑥) = (𝑏‘𝑥))
130	129	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝑎‘𝑚) = (𝑏‘𝑚) ↔ ∀𝑥 ∈ {𝑚} (𝑎‘𝑥) = (𝑏‘𝑥)))
131	124, 130	anbi12d 632	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (((𝑎 ↾ 𝑚) = (𝑏 ↾ 𝑚) ∧ (𝑎‘𝑚) = (𝑏‘𝑚)) ↔ (∀𝑥 ∈ 𝑚 (𝑎‘𝑥) = (𝑏‘𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎‘𝑥) = (𝑏‘𝑥))))
132	93	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝐺‘suc 𝑚) = (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚))))
133	132	fveq1d 6874	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑎) = ((𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))‘𝑎))
134		reseq1 5957	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑎 → (𝑧 ↾ 𝑚) = (𝑎 ↾ 𝑚))
135	134	fveq2d 6876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑎 → ((𝐺‘𝑚)‘(𝑧 ↾ 𝑚)) = ((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)))
136		fveq1 6871	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑎 → (𝑧‘𝑚) = (𝑎‘𝑚))
137	135, 136	oveq12d 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑎 → (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)) = (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚))𝐹(𝑎‘𝑚)))
138		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚))) = (𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))
139		ovex 7432	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚))𝐹(𝑎‘𝑚)) ∈ V
140	137, 138, 139	fvmpt 6982	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) → ((𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))‘𝑎) = (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚))𝐹(𝑎‘𝑚)))
141	140	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))‘𝑎) = (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚))𝐹(𝑎‘𝑚)))
142	133, 141	eqtrd 2769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑎) = (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚))𝐹(𝑎‘𝑚)))
143		df-ov 7402	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚))𝐹(𝑎‘𝑚)) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩)
144	142, 143	eqtrdi 2785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑎) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩))
145	132	fveq1d 6874	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑏) = ((𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))‘𝑏))
146		reseq1 5957	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑏 → (𝑧 ↾ 𝑚) = (𝑏 ↾ 𝑚))
147	146	fveq2d 6876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑏 → ((𝐺‘𝑚)‘(𝑧 ↾ 𝑚)) = ((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)))
148		fveq1 6871	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑏 → (𝑧‘𝑚) = (𝑏‘𝑚))
149	147, 148	oveq12d 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑏 → (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)) = (((𝐺‘𝑚)‘(𝑏 ↾ 𝑚))𝐹(𝑏‘𝑚)))
150		ovex 7432	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺‘𝑚)‘(𝑏 ↾ 𝑚))𝐹(𝑏‘𝑚)) ∈ V
151	149, 138, 150	fvmpt 6982	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (𝐴 ↑_m suc 𝑚) → ((𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))‘𝑏) = (((𝐺‘𝑚)‘(𝑏 ↾ 𝑚))𝐹(𝑏‘𝑚)))
152	151	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝑧 ∈ (𝐴 ↑_m suc 𝑚) ↦ (((𝐺‘𝑚)‘(𝑧 ↾ 𝑚))𝐹(𝑧‘𝑚)))‘𝑏) = (((𝐺‘𝑚)‘(𝑏 ↾ 𝑚))𝐹(𝑏‘𝑚)))
153	145, 152	eqtrd 2769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑏) = (((𝐺‘𝑚)‘(𝑏 ↾ 𝑚))𝐹(𝑏‘𝑚)))
154		df-ov 7402	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑚)‘(𝑏 ↾ 𝑚))𝐹(𝑏‘𝑚)) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩)
155	153, 154	eqtrdi 2785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘suc 𝑚)‘𝑏) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩))
156	144, 155	eqeq12d 2750	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) ↔ (𝐹‘⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩)))
157	94	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
158		f1of1 6813	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴 → 𝐹:(𝐴 × 𝐴)–1-1→𝐴)
159	157, 158	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝐹:(𝐴 × 𝐴)–1-1→𝐴)
160	99	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝐺‘𝑚):(𝐴 ↑_m 𝑚)⟶𝐴)
161		fssres 6740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎:suc 𝑚⟶𝐴 ∧ 𝑚 ⊆ suc 𝑚) → (𝑎 ↾ 𝑚):𝑚⟶𝐴)
162	117, 103, 161	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑎 ↾ 𝑚):𝑚⟶𝐴)
163	44	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → 𝐴 ∈ 𝑉)
164		elmapg 8847	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑚 ∈ V) → ((𝑎 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚) ↔ (𝑎 ↾ 𝑚):𝑚⟶𝐴))
165	163, 54, 164	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝑎 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚) ↔ (𝑎 ↾ 𝑚):𝑚⟶𝐴))
166	162, 165	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑎 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚))
167	160, 166	ffvelcdmd 7071	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)) ∈ 𝐴)
168		ffvelcdm 7067	. . . . . . . . . . . . . . 15 ⊢ ((𝑎:suc 𝑚⟶𝐴 ∧ 𝑚 ∈ suc 𝑚) → (𝑎‘𝑚) ∈ 𝐴)
169	117, 111, 168	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑎‘𝑚) ∈ 𝐴)
170	167, 169	opelxpd 5690	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩ ∈ (𝐴 × 𝐴))
171		fssres 6740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏:suc 𝑚⟶𝐴 ∧ 𝑚 ⊆ suc 𝑚) → (𝑏 ↾ 𝑚):𝑚⟶𝐴)
172	120, 103, 171	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑏 ↾ 𝑚):𝑚⟶𝐴)
173		elmapg 8847	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑚 ∈ V) → ((𝑏 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚) ↔ (𝑏 ↾ 𝑚):𝑚⟶𝐴))
174	163, 54, 173	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝑏 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚) ↔ (𝑏 ↾ 𝑚):𝑚⟶𝐴))
175	172, 174	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑏 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚))
176	160, 175	ffvelcdmd 7071	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)) ∈ 𝐴)
177		ffvelcdm 7067	. . . . . . . . . . . . . . 15 ⊢ ((𝑏:suc 𝑚⟶𝐴 ∧ 𝑚 ∈ suc 𝑚) → (𝑏‘𝑚) ∈ 𝐴)
178	120, 111, 177	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑏‘𝑚) ∈ 𝐴)
179	176, 178	opelxpd 5690	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩ ∈ (𝐴 × 𝐴))
180		f1fveq 7250	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝐴 × 𝐴)–1-1→𝐴 ∧ (⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩ ∈ (𝐴 × 𝐴) ∧ ⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩ ∈ (𝐴 × 𝐴))) → ((𝐹‘⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩) ↔ ⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩ = ⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩))
181	159, 170, 179, 180	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐹‘⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩) ↔ ⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩ = ⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩))
182		fvex 6885	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)) ∈ V
183		fvex 6885	. . . . . . . . . . . . 13 ⊢ (𝑎‘𝑚) ∈ V
184	182, 183	opth 5448	. . . . . . . . . . . 12 ⊢ (⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩ = ⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩ ↔ (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)) = ((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)) ∧ (𝑎‘𝑚) = (𝑏‘𝑚)))
185	181, 184	bitrdi 287	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((𝐹‘⟨((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)), (𝑎‘𝑚)⟩) = (𝐹‘⟨((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)), (𝑏‘𝑚)⟩) ↔ (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)) = ((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)) ∧ (𝑎‘𝑚) = (𝑏‘𝑚))))
186		simplrr 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)
187		f1fveq 7250	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴 ∧ ((𝑎 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚) ∧ (𝑏 ↾ 𝑚) ∈ (𝐴 ↑_m 𝑚))) → (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)) = ((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)) ↔ (𝑎 ↾ 𝑚) = (𝑏 ↾ 𝑚)))
188	186, 166, 175, 187	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)) = ((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)) ↔ (𝑎 ↾ 𝑚) = (𝑏 ↾ 𝑚)))
189	188	anbi1d 631	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → ((((𝐺‘𝑚)‘(𝑎 ↾ 𝑚)) = ((𝐺‘𝑚)‘(𝑏 ↾ 𝑚)) ∧ (𝑎‘𝑚) = (𝑏‘𝑚)) ↔ ((𝑎 ↾ 𝑚) = (𝑏 ↾ 𝑚) ∧ (𝑎‘𝑚) = (𝑏‘𝑚))))
190	156, 185, 189	3bitrd 305	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) ↔ ((𝑎 ↾ 𝑚) = (𝑏 ↾ 𝑚) ∧ (𝑎‘𝑚) = (𝑏‘𝑚))))
191		eqfnfv 7017	. . . . . . . . . . . 12 ⊢ ((𝑎 Fn suc 𝑚 ∧ 𝑏 Fn suc 𝑚) → (𝑎 = 𝑏 ↔ ∀𝑥 ∈ suc 𝑚(𝑎‘𝑥) = (𝑏‘𝑥)))
192	118, 121, 191	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑎 = 𝑏 ↔ ∀𝑥 ∈ suc 𝑚(𝑎‘𝑥) = (𝑏‘𝑥)))
193		df-suc 6355	. . . . . . . . . . . . 13 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
194	193	raleqi 3301	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ suc 𝑚(𝑎‘𝑥) = (𝑏‘𝑥) ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑚})(𝑎‘𝑥) = (𝑏‘𝑥))
195		ralunb 4170	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑚 ∪ {𝑚})(𝑎‘𝑥) = (𝑏‘𝑥) ↔ (∀𝑥 ∈ 𝑚 (𝑎‘𝑥) = (𝑏‘𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎‘𝑥) = (𝑏‘𝑥)))
196	194, 195	bitri 275	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ suc 𝑚(𝑎‘𝑥) = (𝑏‘𝑥) ↔ (∀𝑥 ∈ 𝑚 (𝑎‘𝑥) = (𝑏‘𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎‘𝑥) = (𝑏‘𝑥)))
197	192, 196	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (𝑎 = 𝑏 ↔ (∀𝑥 ∈ 𝑚 (𝑎‘𝑥) = (𝑏‘𝑥) ∧ ∀𝑥 ∈ {𝑚} (𝑎‘𝑥) = (𝑏‘𝑥))))
198	131, 190, 197	3bitr4d 311	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) ↔ 𝑎 = 𝑏))
199	198	biimpd 229	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) ∧ (𝑎 ∈ (𝐴 ↑_m suc 𝑚) ∧ 𝑏 ∈ (𝐴 ↑_m suc 𝑚))) → (((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) → 𝑎 = 𝑏))
200	199	ralrimivva 3185	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) → ∀𝑎 ∈ (𝐴 ↑_m suc 𝑚)∀𝑏 ∈ (𝐴 ↑_m suc 𝑚)(((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) → 𝑎 = 𝑏))
201		dff13 7243	. . . . . . 7 ⊢ ((𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)–1-1→𝐴 ↔ ((𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)⟶𝐴 ∧ ∀𝑎 ∈ (𝐴 ↑_m suc 𝑚)∀𝑏 ∈ (𝐴 ↑_m suc 𝑚)(((𝐺‘suc 𝑚)‘𝑎) = ((𝐺‘suc 𝑚)‘𝑏) → 𝑎 = 𝑏)))
202	115, 200, 201	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ω ∧ (𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴)) → (𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)–1-1→𝐴)
203	202	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → ((𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴 → (𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)–1-1→𝐴))
204	203	expcom 413	. . . 4 ⊢ (𝑚 ∈ ω → (𝜑 → ((𝐺‘𝑚):(𝐴 ↑_m 𝑚)–1-1→𝐴 → (𝐺‘suc 𝑚):(𝐴 ↑_m suc 𝑚)–1-1→𝐴)))
205	20, 27, 34, 51, 204	finds2 7888	. . 3 ⊢ (𝑦 ∈ ω → (𝜑 → (𝐺‘𝑦):(𝐴 ↑_m 𝑦)–1-1→𝐴))
206	8, 205	vtoclga 3554	. 2 ⊢ (𝐶 ∈ ω → (𝜑 → (𝐺‘𝐶):(𝐴 ↑_m 𝐶)–1-1→𝐴))
207	206	impcom 407	1 ⊢ ((𝜑 ∧ 𝐶 ∈ ω) → (𝐺‘𝐶):(𝐴 ↑_m 𝐶)–1-1→𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3050 Vcvv 3457 ∪ cun 3922 ⊆ wss 3924 ∅c0 4306 {csn 4599 ⟨cop 4605 ↦ cmpt 5198 × cxp 5649 ↾ cres 5653 suc csuc 6351 Fn wfn 6522 ⟶wf 6523 –1-1→wf1 6524 –1-1-onto→wf1o 6526 ‘cfv 6527 (class class class)co 7399 ∈ cmpo 7401 ωcom 7855 seq_ωcseqom 8455 1_oc1o 8467 ↑_m cmap 8834

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5246 ax-sep 5263 ax-nul 5273 ax-pow 5332 ax-pr 5399 ax-un 7723

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4881 df-iun 4966 df-br 5117 df-opab 5179 df-mpt 5199 df-tr 5227 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6287 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6480 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-ov 7402 df-oprab 7403 df-mpo 7404 df-om 7856 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8379 df-rdg 8418 df-seqom 8456 df-1o 8474 df-map 8836

This theorem is referenced by: fseqenlem2 10031

W3C validator