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Theorem dfac12r 10082
Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 10085 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 9729 . . . 4 (𝑦 (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2 harcl 9495 . . . . . . . . 9 (har‘(𝑅1𝑧)) ∈ On
3 pweq 4574 . . . . . . . . . . 11 (𝑥 = (har‘(𝑅1𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1𝑧)))
43eleq1d 2822 . . . . . . . . . 10 (𝑥 = (har‘(𝑅1𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1𝑧)) ∈ dom card))
54rspcv 3577 . . . . . . . . 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 9889 . . . . . . . 8 (𝒫 (har‘(𝑅1𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)))
8 ensym 8943 . . . . . . . 8 ((card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)) → 𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))))
9 bren 8893 . . . . . . . . 9 (𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))) ↔ ∃𝑓 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))))
10 simpr 485 . . . . . . . . . . . 12 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
11 f1of1 6783 . . . . . . . . . . . . . 14 (𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→(card‘𝒫 (har‘(𝑅1𝑧))))
1211adantr 481 . . . . . . . . . . . . 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 9880 . . . . . . . . . . . . . 14 (card‘𝒫 (har‘(𝑅1𝑧))) ∈ On
1413onssi 7773 . . . . . . . . . . . . 13 (card‘𝒫 (har‘(𝑅1𝑧))) ⊆ On
15 f1ss 6744 . . . . . . . . . . . . 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 586 . . . . . . . . . . . 12 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→On)
17 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
1817oveq2d 7373 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (suc ran ran 𝑥 ·o (rank‘𝑦)) = (suc ran ran 𝑥 ·o (rank‘𝑏)))
19 suceq 6383 . . . . . . . . . . . . . . . . . . . . 21 ((rank‘𝑦) = (rank‘𝑏) → suc (rank‘𝑦) = suc (rank‘𝑏))
2017, 19syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → suc (rank‘𝑦) = suc (rank‘𝑏))
2120fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏𝑦 = 𝑏)
2321, 22fveq12d 6849 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
2418, 23oveq12d 7375 . . . . . . . . . . . . . . . . 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 6009 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏))
2625fveq2d 6846 . . . . . . . . . . . . . . . . 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 4507 . . . . . . . . . . . . . . . 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 5218 . . . . . . . . . . . . . . 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 5859 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
3029fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
3129unieqd 4879 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 dom 𝑥 = dom 𝑎)
3229, 31eqeq12d 2752 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (dom 𝑥 = dom 𝑥 ↔ dom 𝑎 = dom 𝑎))
33 rneq 5891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
3433unieqd 4879 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 ran 𝑥 = ran 𝑎)
3534rneqd 5893 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ran ran 𝑥 = ran ran 𝑎)
3635unieqd 4879 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 ran ran 𝑥 = ran ran 𝑎)
37 suceq 6383 . . . . . . . . . . . . . . . . . . . 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 7372 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (suc ran ran 𝑥 ·o (rank‘𝑏)) = (suc ran ran 𝑎 ·o (rank‘𝑏)))
40 fveq1 6841 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
4140fveq1d 6844 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
4239, 41oveq12d 7375 . . . . . . . . . . . . . . . . 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 6849 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 dom 𝑥) = (𝑎 dom 𝑎))
4544rneqd 5893 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → ran (𝑥 dom 𝑥) = ran (𝑎 dom 𝑎))
46 oieq2 9449 . . . . . . . . . . . . . . . . . . . . . 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 5831 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran (𝑎 dom 𝑎)))
4948, 44coeq12d 5820 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)))
5049imaeq1d 6012 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏) = ((OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)) “ 𝑏))
5150fveq2d 6846 . . . . . . . . . . . . . . . . 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 4514 . . . . . . . . . . . . . . . 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 5196 . . . . . . . . . . . . . . 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, 53eqtrid 2788 . . . . . . . . . . . . . 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 5218 . . . . . . . . . . . . 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 8320 . . . . . . . . . . . . 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 10081 . . . . . . . . . . 11 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
5958ex 413 . . . . . . . . . 10 (𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
6059exlimiv 1933 . . . . . . . . 9 (∃𝑓 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
619, 60sylbi 216 . . . . . . . 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 407 . . . . . 6 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
64 r1suc 9706 . . . . . . . . 9 (𝑧 ∈ On → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1𝑧))
6564adantl 482 . . . . . . . 8 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1𝑧))
6665eleq2d 2823 . . . . . . 7 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1𝑧)))
67 elpwi 4567 . . . . . . 7 (𝑦 ∈ 𝒫 (𝑅1𝑧) → 𝑦 ⊆ (𝑅1𝑧))
6866, 67syl6bi 252 . . . . . 6 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ⊆ (𝑅1𝑧)))
69 ssnum 9975 . . . . . 6 (((𝑅1𝑧) ∈ dom card ∧ 𝑦 ⊆ (𝑅1𝑧)) → 𝑦 ∈ dom card)
7063, 68, 69syl6an 682 . . . . 5 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
7170rexlimdva 3152 . . . 4 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
721, 71biimtrid 241 . . 3 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑦 (𝑅1 “ On) → 𝑦 ∈ dom card))
7372ssrdv 3950 . 2 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑅1 “ On) ⊆ dom card)
74 onwf 9766 . . . . . 6 On ⊆ (𝑅1 “ On)
7574sseli 3940 . . . . 5 (𝑥 ∈ On → 𝑥 (𝑅1 “ On))
76 pwwf 9743 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
7775, 76sylib 217 . . . 4 (𝑥 ∈ On → 𝒫 𝑥 (𝑅1 “ On))
78 ssel 3937 . . . 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 3142 . 2 ( (𝑅1 “ On) ⊆ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
8173, 80impbii 208 1 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ (𝑅1 “ On) ⊆ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  ifcif 4486  𝒫 cpw 4560   cuni 4865   class class class wbr 5105  cmpt 5188   E cep 5536  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  ccom 5637  Oncon0 6317  suc csuc 6319  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  recscrecs 8316   +o coa 8409   ·o comu 8410  cen 8880  OrdIsocoi 9445  harchar 9492  𝑅1cr1 9698  rankcrnk 9699  cardccrd 9871
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-oadd 8416  df-omul 8417  df-er 8648  df-en 8884  df-dom 8885  df-oi 9446  df-har 9493  df-r1 9700  df-rank 9701  df-card 9875
This theorem is referenced by:  dfac12a  10084
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