Theorem dfac12r 9557

Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 9560 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 9206	. . . 4 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2		harcl 9007	. . . . . . . . 9 ⊢ (har‘(𝑅1‘𝑧)) ∈ On
3		pweq 4513	. . . . . . . . . . 11 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1‘𝑧)))
4	3	eleq1d 2874	. . . . . . . . . 10 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card))
5	4	rspcv 3566	. . . . . . . . 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 9366	. . . . . . . 8 ⊢ (𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)))
8		ensym 8541	. . . . . . . 8 ⊢ ((card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)) → 𝒫 (har‘(𝑅1‘𝑧)) ≈ (card‘𝒫 (har‘(𝑅1‘𝑧))))
9		bren 8501	. . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
11		f1of1 6589	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 9357	. . . . . . . . . . . . . 14 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ∈ On
14	13	onssi 7532	. . . . . . . . . . . . 13 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ⊆ On
15		f1ss 6555	. . . . . . . . . . . . 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 589	. . . . . . . . . . . 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 6645	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
18	17	oveq2d 7151	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)))
19		suceq 6224	. . . . . . . . . . . . . . . . . . . . 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 6649	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → 𝑦 = 𝑏)
23	21, 22	fveq12d 6652	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
24	18, 23	oveq12d 7153	. . . . . . . . . . . . . . . . 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 5892	. . . . . . . . . . . . . . . . . 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 6649	. . . . . . . . . . . . . . . . 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 4445	. . . . . . . . . . . . . . . 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 5133	. . . . . . . . . . . . . . 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 5736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
30	29	fveq2d 6649	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
31	29	unieqd 4814	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ∪ dom 𝑥 = ∪ dom 𝑎)
32	29, 31	eqeq12d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (dom 𝑥 = ∪ dom 𝑥 ↔ dom 𝑎 = ∪ dom 𝑎))
33		rneq 5770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
34	33	unieqd 4814	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ∪ ran 𝑥 = ∪ ran 𝑎)
35	34	rneqd 5772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → ran ∪ ran 𝑥 = ran ∪ ran 𝑎)
36	35	unieqd 4814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ ran 𝑎)
37		suceq 6224	. . . . . . . . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) = (suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)))
40		fveq1 6644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
41	40	fveq1d 6647	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
42	39, 41	oveq12d 7153	. . . . . . . . . . . . . . . . 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 6652	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → (𝑥‘∪ dom 𝑥) = (𝑎‘∪ dom 𝑎))
45	44	rneqd 5772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ran (𝑥‘∪ dom 𝑥) = ran (𝑎‘∪ dom 𝑎))
46		oieq2 8961	. . . . . . . . . . . . . . . . . . . . . 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 5710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)))
49	48, 44	coeq12d 5699	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) = (◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)))
50	49	imaeq1d 5895	. . . . . . . . . . . . . . . . . 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 6649	. . . . . . . . . . . . . . . . 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 4452	. . . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . 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	syl5eq 2845	. . . . . . . . . . . . . 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 5133	. . . . . . . . . . . . 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 7993	. . . . . . . . . . . . 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 9556	. . . . . . . . . . 11 ⊢ ((𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) ∧ 𝑧 ∈ On) → (𝑅1‘𝑧) ∈ dom card)
59	58	ex 416	. . . . . . . . . 10 ⊢ (𝑓:𝒫 (har‘(𝑅1‘𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1‘𝑧))) → (𝑧 ∈ On → (𝑅1‘𝑧) ∈ dom card))
60	59	exlimiv 1931	. . . . . . . . 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 220	. . . . . . . 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 410	. . . . . 6 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘𝑧) ∈ dom card)
64		r1suc 9183	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1‘𝑧))
65	64	adantl 485	. . . . . . . 8 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1‘𝑧))
66	65	eleq2d 2875	. . . . . . 7 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1‘𝑧)))
67		elpwi 4506	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (𝑅1‘𝑧) → 𝑦 ⊆ (𝑅1‘𝑧))
68	66, 67	syl6bi 256	. . . . . 6 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ⊆ (𝑅1‘𝑧)))
69		ssnum 9450	. . . . . 6 ⊢ (((𝑅1‘𝑧) ∈ dom card ∧ 𝑦 ⊆ (𝑅1‘𝑧)) → 𝑦 ∈ dom card)
70	63, 68, 69	syl6an 683	. . . . 5 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
71	70	rexlimdva 3243	. . . 4 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
72	1, 71	syl5bi 245	. . 3 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ dom card))
73	72	ssrdv 3921	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∪ (𝑅1 “ On) ⊆ dom card)
74		onwf 9243	. . . . . 6 ⊢ On ⊆ ∪ (𝑅1 “ On)
75	74	sseli 3911	. . . . 5 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ (𝑅1 “ On))
76		pwwf 9220	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
77	75, 76	sylib 221	. . . 4 ⊢ (𝑥 ∈ On → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
78		ssel 3908	. . . 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 3148	. 2 ⊢ (∪ (𝑅1 “ On) ⊆ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
81	73, 80	impbii 212	1 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ifcif 4425  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030   ↦ cmpt 5110   E cep 5429  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522   ∘ ccom 5523  Oncon0 6159  suc csuc 6161  –1-1→wf1 6321  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  recscrecs 7990   +_o coa 8082   ·_o comu 8083   ≈ cen 8489  OrdIsocoi 8957  harchar 9004  𝑅₁cr1 9175  rankcrnk 9176  cardccrd 9348

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-omul 8090  df-er 8272  df-en 8493  df-dom 8494  df-oi 8958  df-har 9005  df-r1 9177  df-rank 9178  df-card 9352

This theorem is referenced by:  dfac12a  9559