Theorem dfac12r 9886

Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 9889 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 9535	. . . 4 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2		harcl 9279	. . . . . . . . 9 ⊢ (har‘(𝑅1‘𝑧)) ∈ On
3		pweq 4554	. . . . . . . . . . 11 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1‘𝑧)))
4	3	eleq1d 2824	. . . . . . . . . 10 ⊢ (𝑥 = (har‘(𝑅1‘𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card))
5	4	rspcv 3555	. . . . . . . . 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 9695	. . . . . . . 8 ⊢ (𝒫 (har‘(𝑅1‘𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)))
8		ensym 8760	. . . . . . . 8 ⊢ ((card‘𝒫 (har‘(𝑅1‘𝑧))) ≈ 𝒫 (har‘(𝑅1‘𝑧)) → 𝒫 (har‘(𝑅1‘𝑧)) ≈ (card‘𝒫 (har‘(𝑅1‘𝑧))))
9		bren 8717	. . . . . . . . 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 6711	. . . . . . . . . . . . . 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 9686	. . . . . . . . . . . . . 14 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ∈ On
14	13	onssi 7672	. . . . . . . . . . . . 13 ⊢ (card‘𝒫 (har‘(𝑅1‘𝑧))) ⊆ On
15		f1ss 6672	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 6768	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
18	17	oveq2d 7284	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)))
19		suceq 6328	. . . . . . . . . . . . . . . . . . . . 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 6772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → 𝑦 = 𝑏)
23	21, 22	fveq12d 6775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
24	18, 23	oveq12d 7286	. . . . . . . . . . . . . . . . 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 5962	. . . . . . . . . . . . . . . . . 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 6772	. . . . . . . . . . . . . . . . 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 4485	. . . . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . . . 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 5809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
30	29	fveq2d 6772	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
31	29	unieqd 4858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ∪ dom 𝑥 = ∪ dom 𝑎)
32	29, 31	eqeq12d 2755	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (dom 𝑥 = ∪ dom 𝑥 ↔ dom 𝑎 = ∪ dom 𝑎))
33		rneq 5842	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
34	33	unieqd 4858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ∪ ran 𝑥 = ∪ ran 𝑎)
35	34	rneqd 5844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → ran ∪ ran 𝑥 = ran ∪ ran 𝑎)
36	35	unieqd 4858	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ ran 𝑎)
37		suceq 6328	. . . . . . . . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑏)) = (suc ∪ ran ∪ ran 𝑎 ·o (rank‘𝑏)))
40		fveq1 6767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
41	40	fveq1d 6770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
42	39, 41	oveq12d 7286	. . . . . . . . . . . . . . . . 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 6775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → (𝑥‘∪ dom 𝑥) = (𝑎‘∪ dom 𝑎))
45	44	rneqd 5844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → ran (𝑥‘∪ dom 𝑥) = ran (𝑎‘∪ dom 𝑎))
46		oieq2 9233	. . . . . . . . . . . . . . . . . . . . . 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 5781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)))
49	48, 44	coeq12d 5770	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) = (◡OrdIso( E , ran (𝑎‘∪ dom 𝑎)) ∘ (𝑎‘∪ dom 𝑎)))
50	49	imaeq1d 5965	. . . . . . . . . . . . . . . . . 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 6772	. . . . . . . . . . . . . . . . 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 4492	. . . . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . 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 8189	. . . . . . . . . . . . 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 9885	. . . . . . . . . . 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 1936	. . . . . . . . 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 406	. . . . . 6 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘𝑧) ∈ dom card)
64		r1suc 9512	. . . . . . . . 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 2825	. . . . . . 7 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1‘𝑧)))
67		elpwi 4547	. . . . . . 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 9779	. . . . . 6 ⊢ (((𝑅1‘𝑧) ∈ dom card ∧ 𝑦 ⊆ (𝑅1‘𝑧)) → 𝑦 ∈ dom card)
70	63, 68, 69	syl6an 680	. . . . 5 ⊢ ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
71	70	rexlimdva 3214	. . . 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 3931	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∪ (𝑅1 “ On) ⊆ dom card)
74		onwf 9572	. . . . . 6 ⊢ On ⊆ ∪ (𝑅1 “ On)
75	74	sseli 3921	. . . . 5 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ (𝑅1 “ On))
76		pwwf 9549	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
77	75, 76	sylib 217	. . . 4 ⊢ (𝑥 ∈ On → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
78		ssel 3918	. . . 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 3108	. 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 395   = wceq 1541  ∃wex 1785   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066  Vcvv 3430   ⊆ wss 3891  ifcif 4464  𝒫 cpw 4538  ∪ cuni 4844   class class class wbr 5078   ↦ cmpt 5161   E cep 5493  ^◡ccnv 5587  dom cdm 5588  ran crn 5589   “ cima 5591   ∘ ccom 5592  Oncon0 6263  suc csuc 6265  –1-1→wf1 6427  –1-1-onto→wf1o 6429  ‘cfv 6430  (class class class)co 7268  recscrecs 8185   +_o coa 8278   ·_o comu 8279   ≈ cen 8704  OrdIsocoi 9229  harchar 9276  𝑅₁cr1 9504  rankcrnk 9505  cardccrd 9677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-oadd 8285  df-omul 8286  df-er 8472  df-en 8708  df-dom 8709  df-oi 9230  df-har 9277  df-r1 9506  df-rank 9507  df-card 9681

This theorem is referenced by:  dfac12a  9888