Theorem ackbij2lem2 10005

Description: Lemma for ackbij2 10008. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypotheses

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
ackbij.g	⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))

Assertion

Ref	Expression
ackbij2lem2	⊢ (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1‘𝐴)–1-1-onto→(card‘(𝑅1‘𝐴)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij2lem2

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6783	. . 3 ⊢ (𝑎 = ∅ → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘∅))
2		fveq2 6783	. . 3 ⊢ (𝑎 = ∅ → (𝑅1‘𝑎) = (𝑅1‘∅))
3		2fveq3 6788	. . 3 ⊢ (𝑎 = ∅ → (card‘(𝑅1‘𝑎)) = (card‘(𝑅1‘∅)))
4	1, 2, 3	f1oeq123d 6719	. 2 ⊢ (𝑎 = ∅ → ((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)) ↔ (rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅))))
5		fveq2 6783	. . 3 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
6		fveq2 6783	. . 3 ⊢ (𝑎 = 𝑏 → (𝑅1‘𝑎) = (𝑅1‘𝑏))
7		2fveq3 6788	. . 3 ⊢ (𝑎 = 𝑏 → (card‘(𝑅1‘𝑎)) = (card‘(𝑅1‘𝑏)))
8	5, 6, 7	f1oeq123d 6719	. 2 ⊢ (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)) ↔ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))))
9		fveq2 6783	. . 3 ⊢ (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
10		fveq2 6783	. . 3 ⊢ (𝑎 = suc 𝑏 → (𝑅1‘𝑎) = (𝑅1‘suc 𝑏))
11		2fveq3 6788	. . 3 ⊢ (𝑎 = suc 𝑏 → (card‘(𝑅1‘𝑎)) = (card‘(𝑅1‘suc 𝑏)))
12	9, 10, 11	f1oeq123d 6719	. 2 ⊢ (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)) ↔ (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
13		fveq2 6783	. . 3 ⊢ (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
14		fveq2 6783	. . 3 ⊢ (𝑎 = 𝐴 → (𝑅1‘𝑎) = (𝑅1‘𝐴))
15		2fveq3 6788	. . 3 ⊢ (𝑎 = 𝐴 → (card‘(𝑅1‘𝑎)) = (card‘(𝑅1‘𝐴)))
16	13, 14, 15	f1oeq123d 6719	. 2 ⊢ (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)) ↔ (rec(𝐺, ∅)‘𝐴):(𝑅1‘𝐴)–1-1-onto→(card‘(𝑅1‘𝐴))))
17		f1o0 6762	. . 3 ⊢ ∅:∅–1-1-onto→∅
18		0ex 5232	. . . . . 6 ⊢ ∅ ∈ V
19	18	rdg0 8261	. . . . 5 ⊢ (rec(𝐺, ∅)‘∅) = ∅
20		f1oeq1 6713	. . . . 5 ⊢ ((rec(𝐺, ∅)‘∅) = ∅ → ((rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅))))
21	19, 20	ax-mp 5	. . . 4 ⊢ ((rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)))
22		r10 9535	. . . . 5 ⊢ (𝑅1‘∅) = ∅
23	22	fveq2i 6786	. . . . . 6 ⊢ (card‘(𝑅1‘∅)) = (card‘∅)
24		card0 9725	. . . . . 6 ⊢ (card‘∅) = ∅
25	23, 24	eqtri 2767	. . . . 5 ⊢ (card‘(𝑅1‘∅)) = ∅
26		f1oeq23 6716	. . . . 5 ⊢ (((𝑅1‘∅) = ∅ ∧ (card‘(𝑅1‘∅)) = ∅) → (∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:∅–1-1-onto→∅))
27	22, 25, 26	mp2an 689	. . . 4 ⊢ (∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:∅–1-1-onto→∅)
28	21, 27	bitri 274	. . 3 ⊢ ((rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:∅–1-1-onto→∅)
29	17, 28	mpbir 230	. 2 ⊢ (rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅))
30		ackbij.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
31	30	ackbij1lem17 10001	. . . . . . . . 9 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
32	31	a1i 11	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝐹:(𝒫 ω ∩ Fin)–1-1→ω)
33		r1fin 9540	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝑅1‘𝑏) ∈ Fin)
34		ficardom 9728	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑏) ∈ Fin → (card‘(𝑅1‘𝑏)) ∈ ω)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (card‘(𝑅1‘𝑏)) ∈ ω)
36		ackbij2lem1 9984	. . . . . . . . 9 ⊢ ((card‘(𝑅1‘𝑏)) ∈ ω → 𝒫 (card‘(𝑅1‘𝑏)) ⊆ (𝒫 ω ∩ Fin))
37	35, 36	syl 17	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝒫 (card‘(𝑅1‘𝑏)) ⊆ (𝒫 ω ∩ Fin))
38		f1ores 6739	. . . . . . . 8 ⊢ ((𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ 𝒫 (card‘(𝑅1‘𝑏)) ⊆ (𝒫 ω ∩ Fin)) → (𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))):𝒫 (card‘(𝑅1‘𝑏))–1-1-onto→(𝐹 “ 𝒫 (card‘(𝑅1‘𝑏))))
39	32, 37, 38	syl2anc 584	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))):𝒫 (card‘(𝑅1‘𝑏))–1-1-onto→(𝐹 “ 𝒫 (card‘(𝑅1‘𝑏))))
40	30	ackbij1b 10004	. . . . . . . . . 10 ⊢ ((card‘(𝑅1‘𝑏)) ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅1‘𝑏))) = (card‘𝒫 (card‘(𝑅1‘𝑏))))
41	35, 40	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅1‘𝑏))) = (card‘𝒫 (card‘(𝑅1‘𝑏))))
42		ficardid 9729	. . . . . . . . . 10 ⊢ ((𝑅1‘𝑏) ∈ Fin → (card‘(𝑅1‘𝑏)) ≈ (𝑅1‘𝑏))
43		pwen 8946	. . . . . . . . . 10 ⊢ ((card‘(𝑅1‘𝑏)) ≈ (𝑅1‘𝑏) → 𝒫 (card‘(𝑅1‘𝑏)) ≈ 𝒫 (𝑅1‘𝑏))
44		carden2b 9734	. . . . . . . . . 10 ⊢ (𝒫 (card‘(𝑅1‘𝑏)) ≈ 𝒫 (𝑅1‘𝑏) → (card‘𝒫 (card‘(𝑅1‘𝑏))) = (card‘𝒫 (𝑅1‘𝑏)))
45	33, 42, 43, 44	4syl 19	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (card‘𝒫 (card‘(𝑅1‘𝑏))) = (card‘𝒫 (𝑅1‘𝑏)))
46	41, 45	eqtrd 2779	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅1‘𝑏))) = (card‘𝒫 (𝑅1‘𝑏)))
47	46	f1oeq3d 6722	. . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))):𝒫 (card‘(𝑅1‘𝑏))–1-1-onto→(𝐹 “ 𝒫 (card‘(𝑅1‘𝑏))) ↔ (𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))):𝒫 (card‘(𝑅1‘𝑏))–1-1-onto→(card‘𝒫 (𝑅1‘𝑏))))
48	39, 47	mpbid 231	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))):𝒫 (card‘(𝑅1‘𝑏))–1-1-onto→(card‘𝒫 (𝑅1‘𝑏)))
49	48	adantr 481	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))):𝒫 (card‘(𝑅1‘𝑏))–1-1-onto→(card‘𝒫 (𝑅1‘𝑏)))
50		f1opw 7534	. . . . . 6 ⊢ ((rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏)) → (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1‘𝑏)–1-1-onto→𝒫 (card‘(𝑅1‘𝑏)))
51	50	adantl 482	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1‘𝑏)–1-1-onto→𝒫 (card‘(𝑅1‘𝑏)))
52		f1oco 6748	. . . . 5 ⊢ (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))):𝒫 (card‘(𝑅1‘𝑏))–1-1-onto→(card‘𝒫 (𝑅1‘𝑏)) ∧ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1‘𝑏)–1-1-onto→𝒫 (card‘(𝑅1‘𝑏))) → ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1‘𝑏)–1-1-onto→(card‘𝒫 (𝑅1‘𝑏)))
53	49, 51, 52	syl2anc 584	. . . 4 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1‘𝑏)–1-1-onto→(card‘𝒫 (𝑅1‘𝑏)))
54		frsuc 8277	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)))
55		peano2 7746	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
56	55	fvresd 6803	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (rec(𝐺, ∅)‘suc 𝑏))
57		fvres 6802	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘𝑏) = (rec(𝐺, ∅)‘𝑏))
58	57	fveq2d 6787	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
59		fvex 6796	. . . . . . . . . . 11 ⊢ (rec(𝐺, ∅)‘𝑏) ∈ V
60		dmeq 5815	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → dom 𝑥 = dom (rec(𝐺, ∅)‘𝑏))
61	60	pweqd 4553	. . . . . . . . . . . . 13 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → 𝒫 dom 𝑥 = 𝒫 dom (rec(𝐺, ∅)‘𝑏))
62		imaeq1 5967	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝑥 “ 𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑦))
63	62	fveq2d 6787	. . . . . . . . . . . . 13 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝐹‘(𝑥 “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
64	61, 63	mpteq12dv 5166	. . . . . . . . . . . 12 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
65		ackbij.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))
66	59	dmex 7767	. . . . . . . . . . . . . 14 ⊢ dom (rec(𝐺, ∅)‘𝑏) ∈ V
67	66	pwex 5304	. . . . . . . . . . . . 13 ⊢ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ∈ V
68	67	mptex 7108	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) ∈ V
69	64, 65, 68	fvmpt 6884	. . . . . . . . . . 11 ⊢ ((rec(𝐺, ∅)‘𝑏) ∈ V → (𝐺‘(rec(𝐺, ∅)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
70	59, 69	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐺‘(rec(𝐺, ∅)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
71	58, 70	eqtrdi 2795	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
72	54, 56, 71	3eqtr3d 2787	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
73	72	adantr 481	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
74		f1odm 6729	. . . . . . . . . . 11 ⊢ ((rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏)) → dom (rec(𝐺, ∅)‘𝑏) = (𝑅1‘𝑏))
75	74	adantl 482	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → dom (rec(𝐺, ∅)‘𝑏) = (𝑅1‘𝑏))
76	75	pweqd 4553	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → 𝒫 dom (rec(𝐺, ∅)‘𝑏) = 𝒫 (𝑅1‘𝑏))
77	76	mpteq1d 5170	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = (𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
78		fvex 6796	. . . . . . . . . . 11 ⊢ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) ∈ V
79		eqid 2739	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = (𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
80	78, 79	fnmpti 6585	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) Fn 𝒫 (𝑅1‘𝑏)
81	80	a1i 11	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) Fn 𝒫 (𝑅1‘𝑏))
82		f1ofn 6726	. . . . . . . . . 10 ⊢ (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1‘𝑏)–1-1-onto→(card‘𝒫 (𝑅1‘𝑏)) → ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))) Fn 𝒫 (𝑅1‘𝑏))
83	53, 82	syl 17	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))) Fn 𝒫 (𝑅1‘𝑏))
84		f1of 6725	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1‘𝑏)–1-1-onto→𝒫 (card‘(𝑅1‘𝑏)) → (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1‘𝑏)⟶𝒫 (card‘(𝑅1‘𝑏)))
85	51, 84	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1‘𝑏)⟶𝒫 (card‘(𝑅1‘𝑏)))
86	85	ffvelrnda 6970	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → ((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐) ∈ 𝒫 (card‘(𝑅1‘𝑏)))
87	86	fvresd 6803	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)) = (𝐹‘((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)))
88		imaeq2 5968	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑎) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
89		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)) = (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))
90	59	imaex 7772	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑏) “ 𝑐) ∈ V
91	88, 89, 90	fvmpt 6884	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝒫 (𝑅1‘𝑏) → ((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
92	91	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → ((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
93	92	fveq2d 6787	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → (𝐹‘((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
94	87, 93	eqtrd 2779	. . . . . . . . . 10 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
95		fvco3 6876	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1‘𝑏)⟶𝒫 (card‘(𝑅1‘𝑏)) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)))‘𝑐) = ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)))
96	85, 95	sylan 580	. . . . . . . . . 10 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)))‘𝑐) = ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)))
97		imaeq2 5968	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
98	97	fveq2d 6787	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑐 → (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
99		fvex 6796	. . . . . . . . . . . 12 ⊢ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)) ∈ V
100	98, 79, 99	fvmpt 6884	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (𝑅1‘𝑏) → ((𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
101	100	adantl 482	. . . . . . . . . 10 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → ((𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
102	94, 96, 101	3eqtr4rd 2790	. . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1‘𝑏)) → ((𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))‘𝑐) = (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)))‘𝑐))
103	81, 83, 102	eqfnfvd 6921	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (𝑦 ∈ 𝒫 (𝑅1‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))))
104	77, 103	eqtrd 2779	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))))
105	73, 104	eqtrd 2779	. . . . . 6 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) = ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))))
106		f1oeq1 6713	. . . . . 6 ⊢ ((rec(𝐺, ∅)‘suc 𝑏) = ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))) → ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
107	105, 106	syl 17	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
108		nnon 7727	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
109		r1suc 9537	. . . . . . . 8 ⊢ (𝑏 ∈ On → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1‘𝑏))
110	108, 109	syl 17	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1‘𝑏))
111	110	fveq2d 6787	. . . . . . 7 ⊢ (𝑏 ∈ ω → (card‘(𝑅1‘suc 𝑏)) = (card‘𝒫 (𝑅1‘𝑏)))
112		f1oeq23 6716	. . . . . . 7 ⊢ (((𝑅1‘suc 𝑏) = 𝒫 (𝑅1‘𝑏) ∧ (card‘(𝑅1‘suc 𝑏)) = (card‘𝒫 (𝑅1‘𝑏))) → (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1‘𝑏)–1-1-onto→(card‘𝒫 (𝑅1‘𝑏))))
113	110, 111, 112	syl2anc 584	. . . . . 6 ⊢ (𝑏 ∈ ω → (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1‘𝑏)–1-1-onto→(card‘𝒫 (𝑅1‘𝑏))))
114	113	adantr 481	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1‘𝑏)–1-1-onto→(card‘𝒫 (𝑅1‘𝑏))))
115	107, 114	bitrd 278	. . . 4 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1‘𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1‘𝑏)–1-1-onto→(card‘𝒫 (𝑅1‘𝑏))))
116	53, 115	mpbird 256	. . 3 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏))) → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
117	116	ex 413	. 2 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅)‘𝑏):(𝑅1‘𝑏)–1-1-onto→(card‘(𝑅1‘𝑏)) → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
118	4, 8, 12, 16, 29, 117	finds 7754	1 ⊢ (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1‘𝐴)–1-1-onto→(card‘(𝑅1‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2107  Vcvv 3433   ∩ cin 3887   ⊆ wss 3888  ∅c0 4257  𝒫 cpw 4534  {csn 4562  ∪ ciun 4925   class class class wbr 5075   ↦ cmpt 5158   × cxp 5588  dom cdm 5590   ↾ cres 5592   “ cima 5593   ∘ ccom 5594  Oncon0 6270  suc csuc 6272   Fn wfn 6432  ⟶wf 6433  –1-1→wf1 6434  –1-1-onto→wf1o 6436  ‘cfv 6437  ωcom 7721  reccrdg 8249   ≈ cen 8739  Fincfn 8742  𝑅₁cr1 9529  cardccrd 9702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-2o 8307  df-oadd 8310  df-er 8507  df-map 8626  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-r1 9531  df-dju 9668  df-card 9706

This theorem is referenced by:  ackbij2lem3  10006  ackbij2  10008