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Theorem ackbij2lem2 10258

Description: Lemma for ackbij2 10261. (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 6890	. . 3 ⊢ (𝑎 = ∅ → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘∅))
2		fveq2 6890	. . 3 ⊢ (𝑎 = ∅ → (𝑅₁‘𝑎) = (𝑅₁‘∅))
3		2fveq3 6895	. . 3 ⊢ (𝑎 = ∅ → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘∅)))
4	1, 2, 3	f1oeq123d 6826	. 2 ⊢ (𝑎 = ∅ → ((rec(𝐺, ∅)‘𝑎):(𝑅₁‘𝑎)–1-1-onto→(card‘(𝑅₁‘𝑎)) ↔ (rec(𝐺, ∅)‘∅):(𝑅₁‘∅)–1-1-onto→(card‘(𝑅₁‘∅))))
5		fveq2 6890	. . 3 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
6		fveq2 6890	. . 3 ⊢ (𝑎 = 𝑏 → (𝑅₁‘𝑎) = (𝑅₁‘𝑏))
7		2fveq3 6895	. . 3 ⊢ (𝑎 = 𝑏 → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘𝑏)))
8	5, 6, 7	f1oeq123d 6826	. 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 6890	. . 3 ⊢ (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
10		fveq2 6890	. . 3 ⊢ (𝑎 = suc 𝑏 → (𝑅₁‘𝑎) = (𝑅₁‘suc 𝑏))
11		2fveq3 6895	. . 3 ⊢ (𝑎 = suc 𝑏 → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘suc 𝑏)))
12	9, 10, 11	f1oeq123d 6826	. 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 6890	. . 3 ⊢ (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
14		fveq2 6890	. . 3 ⊢ (𝑎 = 𝐴 → (𝑅₁‘𝑎) = (𝑅₁‘𝐴))
15		2fveq3 6895	. . 3 ⊢ (𝑎 = 𝐴 → (card‘(𝑅₁‘𝑎)) = (card‘(𝑅₁‘𝐴)))
16	13, 14, 15	f1oeq123d 6826	. 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 6869	. . 3 ⊢ ∅:∅–1-1-onto→∅
18		0ex 5303	. . . . . 6 ⊢ ∅ ∈ V
19	18	rdg0 8435	. . . . 5 ⊢ (rec(𝐺, ∅)‘∅) = ∅
20		f1oeq1 6820	. . . . 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 9786	. . . . 5 ⊢ (𝑅₁‘∅) = ∅
23	22	fveq2i 6893	. . . . . 6 ⊢ (card‘(𝑅₁‘∅)) = (card‘∅)
24		card0 9976	. . . . . 6 ⊢ (card‘∅) = ∅
25	23, 24	eqtri 2753	. . . . 5 ⊢ (card‘(𝑅₁‘∅)) = ∅
26		f1oeq23 6823	. . . . 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 10254	. . . . . . . . 9 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
32	31	a1i 11	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝐹:(𝒫 ω ∩ Fin)–1-1→ω)
33		r1fin 9791	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝑅₁‘𝑏) ∈ Fin)
34		ficardom 9979	. . . . . . . . . 10 ⊢ ((𝑅₁‘𝑏) ∈ Fin → (card‘(𝑅₁‘𝑏)) ∈ ω)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (card‘(𝑅₁‘𝑏)) ∈ ω)
36		ackbij2lem1 10237	. . . . . . . . 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 6846	. . . . . . . 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 582	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝐹 ↾ 𝒫 (card‘(𝑅₁‘𝑏))):𝒫 (card‘(𝑅₁‘𝑏))–1-1-onto→(𝐹 “ 𝒫 (card‘(𝑅₁‘𝑏))))
40	30	ackbij1b 10257	. . . . . . . . . 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 9980	. . . . . . . . . 10 ⊢ ((𝑅₁‘𝑏) ∈ Fin → (card‘(𝑅₁‘𝑏)) ≈ (𝑅₁‘𝑏))
43		pwen 9168	. . . . . . . . . 10 ⊢ ((card‘(𝑅₁‘𝑏)) ≈ (𝑅₁‘𝑏) → 𝒫 (card‘(𝑅₁‘𝑏)) ≈ 𝒫 (𝑅₁‘𝑏))
44		carden2b 9985	. . . . . . . . . 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 2765	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅₁‘𝑏))) = (card‘𝒫 (𝑅₁‘𝑏)))
47	46	f1oeq3d 6829	. . . . . . 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 479	. . . . 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 7671	. . . . . 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 480	. . . . 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 6855	. . . . 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 582	. . . 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 8451	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)))
55		peano2 7891	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
56	55	fvresd 6910	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (rec(𝐺, ∅)‘suc 𝑏))
57		fvres 6909	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘𝑏) = (rec(𝐺, ∅)‘𝑏))
58	57	fveq2d 6894	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
59		fvex 6903	. . . . . . . . . . 11 ⊢ (rec(𝐺, ∅)‘𝑏) ∈ V
60		dmeq 5901	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → dom 𝑥 = dom (rec(𝐺, ∅)‘𝑏))
61	60	pweqd 4616	. . . . . . . . . . . . 13 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → 𝒫 dom 𝑥 = 𝒫 dom (rec(𝐺, ∅)‘𝑏))
62		imaeq1 6054	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝑥 “ 𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑦))
63	62	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝐹‘(𝑥 “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
64	61, 63	mpteq12dv 5235	. . . . . . . . . . . 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 7911	. . . . . . . . . . . . . 14 ⊢ dom (rec(𝐺, ∅)‘𝑏) ∈ V
67	66	pwex 5375	. . . . . . . . . . . . 13 ⊢ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ∈ V
68	67	mptex 7229	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) ∈ V
69	64, 65, 68	fvmpt 6998	. . . . . . . . . . 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 2781	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
72	54, 56, 71	3eqtr3d 2773	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
73	72	adantr 479	. . . . . . 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 6836	. . . . . . . . . . 11 ⊢ ((rec(𝐺, ∅)‘𝑏):(𝑅₁‘𝑏)–1-1-onto→(card‘(𝑅₁‘𝑏)) → dom (rec(𝐺, ∅)‘𝑏) = (𝑅₁‘𝑏))
75	74	adantl 480	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅₁‘𝑏)–1-1-onto→(card‘(𝑅₁‘𝑏))) → dom (rec(𝐺, ∅)‘𝑏) = (𝑅₁‘𝑏))
76	75	pweqd 4616	. . . . . . . . 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 5239	. . . . . . . 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 6903	. . . . . . . . . . 11 ⊢ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) ∈ V
79		eqid 2725	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 (𝑅₁‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = (𝑦 ∈ 𝒫 (𝑅₁‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
80	78, 79	fnmpti 6693	. . . . . . . . . 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 6833	. . . . . . . . . 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 6832	. . . . . . . . . . . . . 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 7087	. . . . . . . . . . . 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 6910	. . . . . . . . . . 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 6055	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑎) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
89		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝒫 (𝑅₁‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)) = (𝑎 ∈ 𝒫 (𝑅₁‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))
90	59	imaex 7916	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑏) “ 𝑐) ∈ V
91	88, 89, 90	fvmpt 6998	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝒫 (𝑅₁‘𝑏) → ((𝑎 ∈ 𝒫 (𝑅₁‘𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
92	91	adantl 480	. . . . . . . . . . . 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 6894	. . . . . . . . . . 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 2765	. . . . . . . . . 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 6990	. . . . . . . . . . 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 578	. . . . . . . . . 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 6055	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
98	97	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑐 → (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
99		fvex 6903	. . . . . . . . . . . 12 ⊢ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)) ∈ V
100	98, 79, 99	fvmpt 6998	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (𝑅₁‘𝑏) → ((𝑦 ∈ 𝒫 (𝑅₁‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
101	100	adantl 480	. . . . . . . . . 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 2776	. . . . . . . . 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 7036	. . . . . . . 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 2765	. . . . . . 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 2765	. . . . . 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 6820	. . . . . 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 7871	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
109		r1suc 9788	. . . . . . . 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 6894	. . . . . . 7 ⊢ (𝑏 ∈ ω → (card‘(𝑅₁‘suc 𝑏)) = (card‘𝒫 (𝑅₁‘𝑏)))
112		f1oeq23 6823	. . . . . . 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 582	. . . . . 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 479	. . . . 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 411	. 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 7898	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 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∩ cin 3940 ⊆ wss 3941 ∅c0 4319 𝒫 cpw 4599 {csn 4625 ∪ ciun 4992 class class class wbr 5144 ↦ cmpt 5227 × cxp 5671 dom cdm 5673 ↾ cres 5675 “ cima 5676 ∘ ccom 5677 Oncon0 6365 suc csuc 6367 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 ωcom 7865 reccrdg 8423 ≈ cen 8954 Fincfn 8957 𝑅₁cr1 9780 cardccrd 9953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-r1 9782 df-dju 9919 df-card 9957

This theorem is referenced by: ackbij2lem3 10259 ackbij2 10261

W3C validator