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Theorem dfac12r 9006
Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 9009 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.
StepHypRef Expression
1 rankwflemb 8694 . . . 4 (𝑦 (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2 harcl 8507 . . . . . . . . 9 (har‘(𝑅1𝑧)) ∈ On
3 pweq 4194 . . . . . . . . . . 11 (𝑥 = (har‘(𝑅1𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1𝑧)))
43eleq1d 2715 . . . . . . . . . 10 (𝑥 = (har‘(𝑅1𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1𝑧)) ∈ dom card))
54rspcv 3336 . . . . . . . . 9 ((har‘(𝑅1𝑧)) ∈ On → (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (har‘(𝑅1𝑧)) ∈ dom card))
62, 5ax-mp 5 . . . . . . . 8 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (har‘(𝑅1𝑧)) ∈ dom card)
7 cardid2 8817 . . . . . . . 8 (𝒫 (har‘(𝑅1𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)))
8 ensym 8046 . . . . . . . 8 ((card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)) → 𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))))
9 bren 8006 . . . . . . . . 9 (𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))) ↔ ∃𝑓 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))))
10 simpr 476 . . . . . . . . . . . 12 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
11 f1of1 6174 . . . . . . . . . . . . . 14 (𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→(card‘𝒫 (har‘(𝑅1𝑧))))
1211adantr 480 . . . . . . . . . . . . 13 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→(card‘𝒫 (har‘(𝑅1𝑧))))
13 cardon 8808 . . . . . . . . . . . . . 14 (card‘𝒫 (har‘(𝑅1𝑧))) ∈ On
1413onssi 7079 . . . . . . . . . . . . 13 (card‘𝒫 (har‘(𝑅1𝑧))) ⊆ On
15 f1ss 6144 . . . . . . . . . . . . 13 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ (card‘𝒫 (har‘(𝑅1𝑧))) ⊆ On) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→On)
1612, 14, 15sylancl 695 . . . . . . . . . . . 12 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→On)
17 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
1817oveq2d 6706 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) = (suc ran ran 𝑥 ·𝑜 (rank‘𝑏)))
19 suceq 5828 . . . . . . . . . . . . . . . . . . . . 21 ((rank‘𝑦) = (rank‘𝑏) → suc (rank‘𝑦) = suc (rank‘𝑏))
2017, 19syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → suc (rank‘𝑦) = suc (rank‘𝑏))
2120fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏𝑦 = 𝑏)
2321, 22fveq12d 6235 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
2418, 23oveq12d 6708 . . . . . . . . . . . . . . . . 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 5497 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏))
2625fveq2d 6233 . . . . . . . . . . . . . . . . 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)))
2724, 26ifeq12d 4139 . . . . . . . . . . . . . . . 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))))
2827cbvmptv 4783 . . . . . . . . . . . . . . 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 5356 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
3029fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
3129unieqd 4478 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 dom 𝑥 = dom 𝑎)
3229, 31eqeq12d 2666 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (dom 𝑥 = dom 𝑥 ↔ dom 𝑎 = dom 𝑎))
33 rneq 5383 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
3433unieqd 4478 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 ran 𝑥 = ran 𝑎)
3534rneqd 5385 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ran ran 𝑥 = ran ran 𝑎)
3635unieqd 4478 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 ran ran 𝑥 = ran ran 𝑎)
37 suceq 5828 . . . . . . . . . . . . . . . . . . . 20 ( ran ran 𝑥 = ran ran 𝑎 → suc ran ran 𝑥 = suc ran ran 𝑎)
3836, 37syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → suc ran ran 𝑥 = suc ran ran 𝑎)
3938oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (suc ran ran 𝑥 ·𝑜 (rank‘𝑏)) = (suc ran ran 𝑎 ·𝑜 (rank‘𝑏)))
40 fveq1 6228 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
4140fveq1d 6231 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
4239, 41oveq12d 6708 . . . . . . . . . . . . . . . . 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 = 𝑎𝑥 = 𝑎)
4443, 31fveq12d 6235 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 dom 𝑥) = (𝑎 dom 𝑎))
4544rneqd 5385 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → ran (𝑥 dom 𝑥) = ran (𝑎 dom 𝑎))
46 oieq2 8459 . . . . . . . . . . . . . . . . . . . . . 22 (ran (𝑥 dom 𝑥) = ran (𝑎 dom 𝑎) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran (𝑎 dom 𝑎)))
4745, 46syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran (𝑎 dom 𝑎)))
4847cnveqd 5330 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran (𝑎 dom 𝑎)))
4948, 44coeq12d 5319 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)))
5049imaeq1d 5500 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏) = ((OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)) “ 𝑏))
5150fveq2d 6233 . . . . . . . . . . . . . . . . 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)))
5232, 42, 51ifbieq12d 4146 . . . . . . . . . . . . . . . 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))))
5330, 52mpteq12dv 4766 . . . . . . . . . . . . . . 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)))))
5428, 53syl5eq 2697 . . . . . . . . . . . . . 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)))))
5554cbvmptv 4783 . . . . . . . . . . . . 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 7515 . . . . . . . . . . . . 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)))))))
5755, 56ax-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))))))
5810, 16, 57dfac12lem3 9005 . . . . . . . . . . 11 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
5958ex 449 . . . . . . . . . 10 (𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
6059exlimiv 1898 . . . . . . . . 9 (∃𝑓 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
619, 60sylbi 207 . . . . . . . 8 (𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
626, 7, 8, 614syl 19 . . . . . . 7 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
6362imp 444 . . . . . 6 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
64 r1suc 8671 . . . . . . . . 9 (𝑧 ∈ On → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1𝑧))
6564adantl 481 . . . . . . . 8 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1𝑧))
6665eleq2d 2716 . . . . . . 7 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1𝑧)))
67 elpwi 4201 . . . . . . 7 (𝑦 ∈ 𝒫 (𝑅1𝑧) → 𝑦 ⊆ (𝑅1𝑧))
6866, 67syl6bi 243 . . . . . 6 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ⊆ (𝑅1𝑧)))
69 ssnum 8900 . . . . . 6 (((𝑅1𝑧) ∈ dom card ∧ 𝑦 ⊆ (𝑅1𝑧)) → 𝑦 ∈ dom card)
7063, 68, 69syl6an 567 . . . . 5 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
7170rexlimdva 3060 . . . 4 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
721, 71syl5bi 232 . . 3 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑦 (𝑅1 “ On) → 𝑦 ∈ dom card))
7372ssrdv 3642 . 2 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑅1 “ On) ⊆ dom card)
74 onwf 8731 . . . . . 6 On ⊆ (𝑅1 “ On)
7574sseli 3632 . . . . 5 (𝑥 ∈ On → 𝑥 (𝑅1 “ On))
76 pwwf 8708 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
7775, 76sylib 208 . . . 4 (𝑥 ∈ On → 𝒫 𝑥 (𝑅1 “ On))
78 ssel 3630 . . . 4 ( (𝑅1 “ On) ⊆ dom card → (𝒫 𝑥 (𝑅1 “ On) → 𝒫 𝑥 ∈ dom card))
7977, 78syl5 34 . . 3 ( (𝑅1 “ On) ⊆ dom card → (𝑥 ∈ On → 𝒫 𝑥 ∈ dom card))
8079ralrimiv 2994 . 2 ( (𝑅1 “ On) ⊆ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
8173, 80impbii 199 1 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ (𝑅1 “ On) ⊆ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607  ifcif 4119  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  cmpt 4762   E cep 5057  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  Oncon0 5761  suc csuc 5763  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  recscrecs 7512   +𝑜 coa 7602   ·𝑜 comu 7603  cen 7994  OrdIsocoi 8455  harchar 8502  𝑅1cr1 8663  rankcrnk 8664  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-omul 7610  df-er 7787  df-en 7998  df-dom 7999  df-oi 8456  df-har 8504  df-r1 8665  df-rank 8666  df-card 8803
This theorem is referenced by:  dfac12a  9008
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