ackbij2lem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ackbij2lem2		Structured version Visualization version GIF version

Theorem ackbij2lem2 10231

Description: Lemma for ackbij2 10234. (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 6888	. . 3 ⊢ (𝑎 = ∅ → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘∅))
2		fveq2 6888	. . 3 ⊢ (𝑎 = ∅ → (𝑅₁‘𝑎) = (𝑅₁‘∅))
3		2fveq3 6893	. . 3 ⊢ (𝑎 = ∅ → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘∅)))
4	1, 2, 3	f1oeq123d 6824	. 2 ⊢ (𝑎 = ∅ → ((rec(𝐺, ∅)‘𝑎):(𝑅₁‘𝑎)–1-1-onto→(card‘(𝑅₁‘𝑎)) ↔ (rec(𝐺, ∅)‘∅):(𝑅₁‘∅)–1-1-onto→(card‘(𝑅₁‘∅))))
5		fveq2 6888	. . 3 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
6		fveq2 6888	. . 3 ⊢ (𝑎 = 𝑏 → (𝑅₁‘𝑎) = (𝑅₁‘𝑏))
7		2fveq3 6893	. . 3 ⊢ (𝑎 = 𝑏 → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘𝑏)))
8	5, 6, 7	f1oeq123d 6824	. 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 6888	. . 3 ⊢ (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
10		fveq2 6888	. . 3 ⊢ (𝑎 = suc 𝑏 → (𝑅₁‘𝑎) = (𝑅₁‘suc 𝑏))
11		2fveq3 6893	. . 3 ⊢ (𝑎 = suc 𝑏 → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘suc 𝑏)))
12	9, 10, 11	f1oeq123d 6824	. 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 6888	. . 3 ⊢ (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
14		fveq2 6888	. . 3 ⊢ (𝑎 = 𝐴 → (𝑅₁‘𝑎) = (𝑅₁‘𝐴))
15		2fveq3 6893	. . 3 ⊢ (𝑎 = 𝐴 → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘𝐴)))
16	13, 14, 15	f1oeq123d 6824	. 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 6867	. . 3 ⊢ ∅:∅–1-1-onto→∅
18		0ex 5306	. . . . . 6 ⊢ ∅ ∈ V
19	18	rdg0 8417	. . . . 5 ⊢ (rec(𝐺, ∅)‘∅) = ∅
20		f1oeq1 6818	. . . . 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 9759	. . . . 5 ⊢ (𝑅₁‘∅) = ∅
23	22	fveq2i 6891	. . . . . 6 ⊢ (card‘(𝑅₁‘∅)) = (card‘∅)
24		card0 9949	. . . . . 6 ⊢ (card‘∅) = ∅
25	23, 24	eqtri 2760	. . . . 5 ⊢ (card‘(𝑅₁‘∅)) = ∅
26		f1oeq23 6821	. . . . 5 ⊢ (((𝑅₁‘∅) = ∅ ∧ (card‘(𝑅₁‘∅)) = ∅) → (∅:(𝑅₁‘∅)–1-1-onto→(card‘(𝑅₁‘∅)) ↔ ∅:∅–1-1-onto→∅))
27	22, 25, 26	mp2an 690	. . . 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 10227	. . . . . . . . 9 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
32	31	a1i 11	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝐹:(𝒫 ω ∩ Fin)–1-1→ω)
33		r1fin 9764	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝑅₁‘𝑏) ∈ Fin)
34		ficardom 9952	. . . . . . . . . 10 ⊢ ((𝑅₁‘𝑏) ∈ Fin → (card‘(𝑅₁‘𝑏)) ∈ ω)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (card‘(𝑅₁‘𝑏)) ∈ ω)
36		ackbij2lem1 10210	. . . . . . . . 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 6844	. . . . . . . 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 10230	. . . . . . . . . 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 9953	. . . . . . . . . 10 ⊢ ((𝑅₁‘𝑏) ∈ Fin → (card‘(𝑅₁‘𝑏)) ≈ (𝑅₁‘𝑏))
43		pwen 9146	. . . . . . . . . 10 ⊢ ((card‘(𝑅₁‘𝑏)) ≈ (𝑅₁‘𝑏) → 𝒫 (card‘(𝑅₁‘𝑏)) ≈ 𝒫 (𝑅₁‘𝑏))
44		carden2b 9958	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅₁‘𝑏))) = (card‘𝒫 (𝑅₁‘𝑏)))
47	46	f1oeq3d 6827	. . . . . . 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 7658	. . . . . 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 6853	. . . . 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 8433	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)))
55		peano2 7877	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
56	55	fvresd 6908	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (rec(𝐺, ∅)‘suc 𝑏))
57		fvres 6907	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘𝑏) = (rec(𝐺, ∅)‘𝑏))
58	57	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
59		fvex 6901	. . . . . . . . . . 11 ⊢ (rec(𝐺, ∅)‘𝑏) ∈ V
60		dmeq 5901	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → dom 𝑥 = dom (rec(𝐺, ∅)‘𝑏))
61	60	pweqd 4618	. . . . . . . . . . . . 13 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → 𝒫 dom 𝑥 = 𝒫 dom (rec(𝐺, ∅)‘𝑏))
62		imaeq1 6052	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝑥 “ 𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑦))
63	62	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝐹‘(𝑥 “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
64	61, 63	mpteq12dv 5238	. . . . . . . . . . . 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 7898	. . . . . . . . . . . . . 14 ⊢ dom (rec(𝐺, ∅)‘𝑏) ∈ V
67	66	pwex 5377	. . . . . . . . . . . . 13 ⊢ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ∈ V
68	67	mptex 7221	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) ∈ V
69	64, 65, 68	fvmpt 6995	. . . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
72	54, 56, 71	3eqtr3d 2780	. . . . . . . 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 6834	. . . . . . . . . . 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 4618	. . . . . . . . 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 5242	. . . . . . . 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 6901	. . . . . . . . . . 11 ⊢ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) ∈ V
79		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 (𝑅₁‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = (𝑦 ∈ 𝒫 (𝑅₁‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
80	78, 79	fnmpti 6690	. . . . . . . . . 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 6831	. . . . . . . . . 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 6830	. . . . . . . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . . . 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 6908	. . . . . . . . . . 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 6053	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑎) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
89		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝒫 (𝑅₁‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)) = (𝑎 ∈ 𝒫 (𝑅₁‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))
90	59	imaex 7903	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑏) “ 𝑐) ∈ V
91	88, 89, 90	fvmpt 6995	. . . . . . . . . . . . 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 6892	. . . . . . . . . . 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 2772	. . . . . . . . . 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 6987	. . . . . . . . . . 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 6053	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
98	97	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑐 → (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
99		fvex 6901	. . . . . . . . . . . 12 ⊢ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)) ∈ V
100	98, 79, 99	fvmpt 6995	. . . . . . . . . . 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 2783	. . . . . . . . 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 7032	. . . . . . . 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 2772	. . . . . . 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 2772	. . . . . 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 6818	. . . . . 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 7857	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
109		r1suc 9761	. . . . . . . 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 6892	. . . . . . 7 ⊢ (𝑏 ∈ ω → (card‘(𝑅₁‘suc 𝑏)) = (card‘𝒫 (𝑅₁‘𝑏)))
112		f1oeq23 6821	. . . . . . 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 7885	1 ⊢ (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅₁‘𝐴)–1-1-onto→(card‘(𝑅₁‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {csn 4627 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 dom cdm 5675 ↾ cres 5677 “ cima 5678 ∘ ccom 5679 Oncon0 6361 suc csuc 6363 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 –1-1-onto→wf1o 6539 ‘cfv 6540 ωcom 7851 reccrdg 8405 ≈ cen 8932 Fincfn 8935 𝑅₁cr1 9753 cardccrd 9926

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-r1 9755 df-dju 9892 df-card 9930

This theorem is referenced by: ackbij2lem3 10232 ackbij2 10234

W3C validator