Theorem dfac12r 10187

Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 10190 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 9833	. . . 4 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2		harcl 9599	. . . . . . . . 9 ⊢ (har‘(𝑅1‘𝑧)) ∈ On
3		pweq 4614	. . . . . . . . . . 11 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1‘𝑧)))
4	3	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card))
5	4	rspcv 3618	. . . . . . . . 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 9993	. . . . . . . 8 ⊢ (𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)))
8		ensym 9043	. . . . . . . 8 ⊢ ((card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)) → 𝒫 (har‘(𝑅1‘𝑧)) ≈ (card‘𝒫 (har‘(𝑅1‘𝑧))))
9		bren 8995	. . . . . . . . 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 9984	. . . . . . . . . . . . . 14 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ∈ On
14	13	onssi 7858	. . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)))
19		suceq 6450	. . . . . . . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . 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 6074	. . . . . . . . . . . . . . . . . 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 4547	. . . . . . . . . . . . . . . 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 5255	. . . . . . . . . . . . . . 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 5914	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
30	29	fveq2d 6910	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
31	29	unieqd 4920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ∪ dom 𝑥 = ∪ dom 𝑎)
32	29, 31	eqeq12d 2753	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (dom 𝑥 = ∪ dom 𝑥 ↔ dom 𝑎 = ∪ dom 𝑎))
33		rneq 5947	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
34	33	unieqd 4920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ∪ ran 𝑥 = ∪ ran 𝑎)
35	34	rneqd 5949	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → ran ∪ ran 𝑥 = ran ∪ ran 𝑎)
36	35	unieqd 4920	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ ran 𝑎)
37		suceq 6450	. . . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . 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 5949	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ran (𝑥‘∪ dom 𝑥) = ran (𝑎‘∪ dom 𝑎))
46		oieq2 9553	. . . . . . . . . . . . . . . . . . . . . 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 5886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)))
49	48, 44	coeq12d 5875	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) = (◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)))
50	49	imaeq1d 6077	. . . . . . . . . . . . . . . . . 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 4554	. . . . . . . . . . . . . . . 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 5233	. . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . 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 5255	. . . . . . . . . . . . 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 8414	. . . . . . . . . . . . 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 10186	. . . . . . . . . . 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 1930	. . . . . . . . 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 9810	. . . . . . . . 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 2827	. . . . . . 7 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1‘𝑧)))
67		elpwi 4607	. . . . . . 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 10079	. . . . . 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 3155	. . . 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 3989	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∪ (𝑅1 “ On) ⊆ dom card)
74		onwf 9870	. . . . . 6 ⊢ On ⊆ ∪ (𝑅1 “ On)
75	74	sseli 3979	. . . . 5 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ (𝑅1 “ On))
76		pwwf 9847	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
77	75, 76	sylib 218	. . . 4 ⊢ (𝑥 ∈ On → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
78		ssel 3977	. . . 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 3145	. 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 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ifcif 4525  𝒫 cpw 4600  ∪ cuni 4907   class class class wbr 5143   ↦ cmpt 5225   E cep 5583  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688   ∘ ccom 5689  Oncon0 6384  suc csuc 6386  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  recscrecs 8410   +_o coa 8503   ·_o comu 8504   ≈ cen 8982  OrdIsocoi 9549  harchar 9596  𝑅₁cr1 9802  rankcrnk 9803  cardccrd 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-omul 8511  df-er 8745  df-en 8986  df-dom 8987  df-oi 9550  df-har 9597  df-r1 9804  df-rank 9805  df-card 9979

This theorem is referenced by:  dfac12a  10189