Theorem dfac12r 10184

Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 10187 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
dfac12r	⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card)

Proof of Theorem dfac12r

Dummy variables 𝑎 𝑏 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankwflemb 9830	. . . 4 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2		harcl 9596	. . . . . . . . 9 ⊢ (har‘(𝑅1‘𝑧)) ∈ On
3		pweq 4618	. . . . . . . . . . 11 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1‘𝑧)))
4	3	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card))
5	4	rspcv 3617	. . . . . . . . 9 ⊢ ((har‘(𝑅1‘𝑧)) ∈ On → (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card))
6	2, 5	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card)
7		cardid2 9990	. . . . . . . 8 ⊢ (𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)))
8		ensym 9041	. . . . . . . 8 ⊢ ((card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)) → 𝒫 (har‘(𝑅1‘𝑧)) ≈ (card‘𝒫 (har‘(𝑅1‘𝑧))))
9		bren 8993	. . . . . . . . 9 ⊢ (𝒫 (har‘(𝑅1‘𝑧)) ≈ (card‘𝒫 (har‘(𝑅1‘𝑧))) ↔ ∃𝑓 𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))))
10		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
11		f1of1 6847	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) → 𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1→(card‘𝒫 (har‘(𝑅1‘𝑧))))
12	11	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → 𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1→(card‘𝒫 (har‘(𝑅1‘𝑧))))
13		cardon 9981	. . . . . . . . . . . . . 14 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ∈ On
14	13	onssi 7857	. . . . . . . . . . . . 13 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ⊆ On
15		f1ss 6809	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ (card‘𝒫 (har‘(𝑅1‘𝑧))) ⊆ On) → 𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1→On)
16	12, 14, 15	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → 𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1→On)
17		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
18	17	oveq2d 7446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)))
19		suceq 6451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((rank‘𝑦) = (rank‘𝑏) → suc (rank‘𝑦) = suc (rank‘𝑏))
20	17, 19	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑏 → suc (rank‘𝑦) = suc (rank‘𝑏))
21	20	fveq2d 6910	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → 𝑦 = 𝑏)
23	21, 22	fveq12d 6913	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
24	18, 23	oveq12d 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) +o ((𝑥‘suc (rank‘𝑏))‘𝑏)))
25		imaeq2 6075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦) = ((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏))
26	25	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)) = (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏)))
27	24, 26	ifeq12d 4551	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))) = if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) +o ((𝑥‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏))))
28	27	cbvmptv 5260	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))) = (𝑏 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) +o ((𝑥‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏))))
29		dmeq 5916	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
30	29	fveq2d 6910	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
31	29	unieqd 4924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ∪ dom 𝑥 = ∪ dom 𝑎)
32	29, 31	eqeq12d 2750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (dom 𝑥 = ∪ dom 𝑥 ↔ dom 𝑎 = ∪ dom 𝑎))
33		rneq 5949	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
34	33	unieqd 4924	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ∪ ran 𝑥 = ∪ ran 𝑎)
35	34	rneqd 5951	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → ran ∪ ran 𝑥 = ran ∪ ran 𝑎)
36	35	unieqd 4924	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ ran 𝑎)
37		suceq 6451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ ran ∪ ran 𝑥 = ∪ ran ∪ ran 𝑎 → suc ∪ ran ∪ ran 𝑥 = suc ∪ ran ∪ ran 𝑎)
38	36, 37	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → suc ∪ ran ∪ ran 𝑥 = suc ∪ ran ∪ ran 𝑎)
39	38	oveq1d 7445	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) = (suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)))
40		fveq1 6905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
41	40	fveq1d 6908	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
42	39, 41	oveq12d 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) +o ((𝑥‘suc (rank‘𝑏))‘𝑏)) = ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)))
43		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
44	43, 31	fveq12d 6913	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → (𝑥‘∪ dom 𝑥) = (𝑎‘∪ dom 𝑎))
45	44	rneqd 5951	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ran (𝑥‘∪ dom 𝑥) = ran (𝑎‘∪ dom 𝑎))
46		oieq2 9550	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ran (𝑥‘∪ dom 𝑥) = ran (𝑎‘∪ dom 𝑎) → OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = OrdIso( E , ran (𝑎‘∪ dom 𝑎)))
47	45, 46	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = OrdIso( E , ran (𝑎‘∪ dom 𝑎)))
48	47	cnveqd 5888	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)))
49	48, 44	coeq12d 5877	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) = (◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)))
50	49	imaeq1d 6078	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏) = ((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏))
51	50	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏)) = (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏)))
52	32, 42, 51	ifbieq12d 4558	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) +o ((𝑥‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏))) = if(dom 𝑎 = ∪ dom 𝑎, ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏))))
53	30, 52	mpteq12dv 5238	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑏 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) +o ((𝑥‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑏)))) = (𝑏 ∈ (𝑅1‘dom 𝑎) ↦ if(dom 𝑎 = ∪ dom 𝑎, ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏)))))
54	28, 53	eqtrid 2786	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))) = (𝑏 ∈ (𝑅1‘dom 𝑎) ↦ if(dom 𝑎 = ∪ dom 𝑎, ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏)))))
55	54	cbvmptv 5260	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))) = (𝑎 ∈ V ↦ (𝑏 ∈ (𝑅1‘dom 𝑎) ↦ if(dom 𝑎 = ∪ dom 𝑎, ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏)))))
56		recseq 8412	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))) = (𝑎 ∈ V ↦ (𝑏 ∈ (𝑅1‘dom 𝑎) ↦ if(dom 𝑎 = ∪ dom 𝑎, ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏))))) → recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))))) = recs((𝑎 ∈ V ↦ (𝑏 ∈ (𝑅1‘dom 𝑎) ↦ if(dom 𝑎 = ∪ dom 𝑎, ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏)))))))
57	55, 56	ax-mp 5	. . . . . . . . . . . 12 ⊢ recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝑓‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))))) = recs((𝑎 ∈ V ↦ (𝑏 ∈ (𝑅1‘dom 𝑎) ↦ if(dom 𝑎 = ∪ dom 𝑎, ((suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)) +o ((𝑎‘suc (rank‘𝑏))‘𝑏)), (𝑓‘((◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)) “ 𝑏))))))
58	10, 16, 57	dfac12lem3 10183	. . . . . . . . . . 11 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → (𝑅1‘𝑧) ∈ dom card)
59	58	ex 412	. . . . . . . . . 10 ⊢ (𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) → (𝑧 ∈ On → (𝑅1‘𝑧) ∈ dom card))
60	59	exlimiv 1927	. . . . . . . . 9 ⊢ (∃𝑓 𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) → (𝑧 ∈ On → (𝑅1‘𝑧) ∈ dom card))
61	9, 60	sylbi 217	. . . . . . . 8 ⊢ (𝒫 (har‘(𝑅1‘𝑧)) ≈ (card‘𝒫 (har‘(𝑅1‘𝑧))) → (𝑧 ∈ On → (𝑅1‘𝑧) ∈ dom card))
62	6, 7, 8, 61	4syl 19	. . . . . . 7 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑧 ∈ On → (𝑅1‘𝑧) ∈ dom card))
63	62	imp 406	. . . . . 6 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘𝑧) ∈ dom card)
64		r1suc 9807	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1‘𝑧))
65	64	adantl 481	. . . . . . . 8 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1‘𝑧))
66	65	eleq2d 2824	. . . . . . 7 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1‘𝑧)))
67		elpwi 4611	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (𝑅1‘𝑧) → 𝑦 ⊆ (𝑅1‘𝑧))
68	66, 67	biimtrdi 253	. . . . . 6 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ⊆ (𝑅1‘𝑧)))
69		ssnum 10076	. . . . . 6 ⊢ (((𝑅1‘𝑧) ∈ dom card ∧ 𝑦 ⊆ (𝑅1‘𝑧)) → 𝑦 ∈ dom card)
70	63, 68, 69	syl6an 684	. . . . 5 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
71	70	rexlimdva 3152	. . . 4 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
72	1, 71	biimtrid 242	. . 3 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ dom card))
73	72	ssrdv 4000	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∪ (𝑅1 “ On) ⊆ dom card)
74		onwf 9867	. . . . . 6 ⊢ On ⊆ ∪ (𝑅1 “ On)
75	74	sseli 3990	. . . . 5 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ (𝑅1 “ On))
76		pwwf 9844	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
77	75, 76	sylib 218	. . . 4 ⊢ (𝑥 ∈ On → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
78		ssel 3988	. . . 4 ⊢ (∪ (𝑅1 “ On) ⊆ dom card → (𝒫 𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ dom card))
79	77, 78	syl5 34	. . 3 ⊢ (∪ (𝑅1 “ On) ⊆ dom card → (𝑥 ∈ On → 𝒫 𝑥 ∈ dom card))
80	79	ralrimiv 3142	. 2 ⊢ (∪ (𝑅1 “ On) ⊆ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
81	73, 80	impbii 209	1 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  Vcvv 3477   ⊆ wss 3962  ifcif 4530  𝒫 cpw 4604  ∪ cuni 4911   class class class wbr 5147   ↦ cmpt 5230   E cep 5587  ^◡ccnv 5687  dom cdm 5688  ran crn 5689   “ cima 5691   ∘ ccom 5692  Oncon0 6385  suc csuc 6387  –1-1→wf1 6559  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430  recscrecs 8408   +_o coa 8501   ·_o comu 8502   ≈ cen 8980  OrdIsocoi 9546  harchar 9593  𝑅₁cr1 9799  rankcrnk 9800  cardccrd 9972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508  df-omul 8509  df-er 8743  df-en 8984  df-dom 8985  df-oi 9547  df-har 9594  df-r1 9801  df-rank 9802  df-card 9976

This theorem is referenced by:  dfac12a  10186