Theorem dfac12r 9902

Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 9905 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 9551	. . . 4 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2		harcl 9318	. . . . . . . . 9 ⊢ (har‘(𝑅1‘𝑧)) ∈ On
3		pweq 4549	. . . . . . . . . . 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 3557	. . . . . . . . 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 9711	. . . . . . . 8 ⊢ (𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)))
8		ensym 8789	. . . . . . . 8 ⊢ ((card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)) → 𝒫 (har‘(𝑅1‘𝑧)) ≈ (card‘𝒫 (har‘(𝑅1‘𝑧))))
9		bren 8743	. . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
11		f1of1 6715	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 9702	. . . . . . . . . . . . . 14 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ∈ On
14	13	onssi 7684	. . . . . . . . . . . . 13 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ⊆ On
15		f1ss 6676	. . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
18	17	oveq2d 7291	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)))
19		suceq 6331	. . . . . . . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → 𝑦 = 𝑏)
23	21, 22	fveq12d 6781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
24	18, 23	oveq12d 7293	. . . . . . . . . . . . . . . . 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 5965	. . . . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . 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 4480	. . . . . . . . . . . . . . . 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 5187	. . . . . . . . . . . . . . 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 5812	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
30	29	fveq2d 6778	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
31	29	unieqd 4853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ∪ dom 𝑥 = ∪ dom 𝑎)
32	29, 31	eqeq12d 2754	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (dom 𝑥 = ∪ dom 𝑥 ↔ dom 𝑎 = ∪ dom 𝑎))
33		rneq 5845	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
34	33	unieqd 4853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ∪ ran 𝑥 = ∪ ran 𝑎)
35	34	rneqd 5847	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → ran ∪ ran 𝑥 = ran ∪ ran 𝑎)
36	35	unieqd 4853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ ran 𝑎)
37		suceq 6331	. . . . . . . . . . . . . . . . . . . 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 7290	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) = (suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)))
40		fveq1 6773	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
41	40	fveq1d 6776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
42	39, 41	oveq12d 7293	. . . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → (𝑥‘∪ dom 𝑥) = (𝑎‘∪ dom 𝑎))
45	44	rneqd 5847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ran (𝑥‘∪ dom 𝑥) = ran (𝑎‘∪ dom 𝑎))
46		oieq2 9272	. . . . . . . . . . . . . . . . . . . . . 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 5784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)))
49	48, 44	coeq12d 5773	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) = (◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)))
50	49	imaeq1d 5968	. . . . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . 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 4487	. . . . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . 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 5187	. . . . . . . . . . . . 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 8205	. . . . . . . . . . . . 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 9901	. . . . . . . . . . 11 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → (𝑅1‘𝑧) ∈ dom card)
59	58	ex 413	. . . . . . . . . 10 ⊢ (𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) → (𝑧 ∈ On → (𝑅1‘𝑧) ∈ dom card))
60	59	exlimiv 1933	. . . . . . . . 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 216	. . . . . . . 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 407	. . . . . 6 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘𝑧) ∈ dom card)
64		r1suc 9528	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1‘𝑧))
65	64	adantl 482	. . . . . . . 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 4542	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (𝑅1‘𝑧) → 𝑦 ⊆ (𝑅1‘𝑧))
68	66, 67	syl6bi 252	. . . . . 6 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ⊆ (𝑅1‘𝑧)))
69		ssnum 9795	. . . . . 6 ⊢ (((𝑅1‘𝑧) ∈ dom card ∧ 𝑦 ⊆ (𝑅1‘𝑧)) → 𝑦 ∈ dom card)
70	63, 68, 69	syl6an 681	. . . . 5 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
71	70	rexlimdva 3213	. . . 4 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
72	1, 71	syl5bi 241	. . 3 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ dom card))
73	72	ssrdv 3927	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∪ (𝑅1 “ On) ⊆ dom card)
74		onwf 9588	. . . . . 6 ⊢ On ⊆ ∪ (𝑅1 “ On)
75	74	sseli 3917	. . . . 5 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ (𝑅1 “ On))
76		pwwf 9565	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
77	75, 76	sylib 217	. . . 4 ⊢ (𝑥 ∈ On → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
78		ssel 3914	. . . 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 3102	. 2 ⊢ (∪ (𝑅1 “ On) ⊆ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
81	73, 80	impbii 208	1 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ifcif 4459  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157   E cep 5494  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592   ∘ ccom 5593  Oncon0 6266  suc csuc 6268  –1-1→wf1 6430  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  recscrecs 8201   +_o coa 8294   ·_o comu 8295   ≈ cen 8730  OrdIsocoi 9268  harchar 9315  𝑅₁cr1 9520  rankcrnk 9521  cardccrd 9693

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

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 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301  df-omul 8302  df-er 8498  df-en 8734  df-dom 8735  df-oi 9269  df-har 9316  df-r1 9522  df-rank 9523  df-card 9697

This theorem is referenced by:  dfac12a  9904