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Theorem dfac12r 9221
Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 9224 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 8871 . . . 4 (𝑦 (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2 harcl 8673 . . . . . . . . 9 (har‘(𝑅1𝑧)) ∈ On
3 pweq 4318 . . . . . . . . . . 11 (𝑥 = (har‘(𝑅1𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1𝑧)))
43eleq1d 2829 . . . . . . . . . 10 (𝑥 = (har‘(𝑅1𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1𝑧)) ∈ dom card))
54rspcv 3457 . . . . . . . . 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 9030 . . . . . . . 8 (𝒫 (har‘(𝑅1𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)))
8 ensym 8209 . . . . . . . 8 ((card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)) → 𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))))
9 bren 8169 . . . . . . . . 9 (𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))) ↔ ∃𝑓 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))))
10 simpr 477 . . . . . . . . . . . 12 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
11 f1of1 6319 . . . . . . . . . . . . . 14 (𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→(card‘𝒫 (har‘(𝑅1𝑧))))
1211adantr 472 . . . . . . . . . . . . 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 9021 . . . . . . . . . . . . . 14 (card‘𝒫 (har‘(𝑅1𝑧))) ∈ On
1413onssi 7235 . . . . . . . . . . . . 13 (card‘𝒫 (har‘(𝑅1𝑧))) ⊆ On
15 f1ss 6288 . . . . . . . . . . . . 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 580 . . . . . . . . . . . 12 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→On)
17 fveq2 6375 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
1817oveq2d 6858 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) = (suc ran ran 𝑥 ·𝑜 (rank‘𝑏)))
19 suceq 5973 . . . . . . . . . . . . . . . . . . . . 21 ((rank‘𝑦) = (rank‘𝑏) → suc (rank‘𝑦) = suc (rank‘𝑏))
2017, 19syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → suc (rank‘𝑦) = suc (rank‘𝑏))
2120fveq2d 6379 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏𝑦 = 𝑏)
2321, 22fveq12d 6382 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
2418, 23oveq12d 6860 . . . . . . . . . . . . . . . . 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 5644 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏))
2625fveq2d 6379 . . . . . . . . . . . . . . . . 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 4263 . . . . . . . . . . . . . . . 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 4909 . . . . . . . . . . . . . . 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 5492 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
3029fveq2d 6379 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
3129unieqd 4604 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 dom 𝑥 = dom 𝑎)
3229, 31eqeq12d 2780 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (dom 𝑥 = dom 𝑥 ↔ dom 𝑎 = dom 𝑎))
33 rneq 5519 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
3433unieqd 4604 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 ran 𝑥 = ran 𝑎)
3534rneqd 5521 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ran ran 𝑥 = ran ran 𝑎)
3635unieqd 4604 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 ran ran 𝑥 = ran ran 𝑎)
37 suceq 5973 . . . . . . . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (suc ran ran 𝑥 ·𝑜 (rank‘𝑏)) = (suc ran ran 𝑎 ·𝑜 (rank‘𝑏)))
40 fveq1 6374 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
4140fveq1d 6377 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
4239, 41oveq12d 6860 . . . . . . . . . . . . . . . . 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 6382 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 dom 𝑥) = (𝑎 dom 𝑎))
4544rneqd 5521 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → ran (𝑥 dom 𝑥) = ran (𝑎 dom 𝑎))
46 oieq2 8625 . . . . . . . . . . . . . . . . . . . . . 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 5466 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran (𝑎 dom 𝑎)))
4948, 44coeq12d 5455 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)))
5049imaeq1d 5647 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏) = ((OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)) “ 𝑏))
5150fveq2d 6379 . . . . . . . . . . . . . . . . 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 4270 . . . . . . . . . . . . . . . 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 4892 . . . . . . . . . . . . . . 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 2811 . . . . . . . . . . . . . 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 4909 . . . . . . . . . . . . 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 7674 . . . . . . . . . . . . 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 9220 . . . . . . . . . . 11 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
5958ex 401 . . . . . . . . . 10 (𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
6059exlimiv 2025 . . . . . . . . 9 (∃𝑓 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
619, 60sylbi 208 . . . . . . . 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 395 . . . . . 6 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
64 r1suc 8848 . . . . . . . . 9 (𝑧 ∈ On → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1𝑧))
6564adantl 473 . . . . . . . 8 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1‘suc 𝑧) = 𝒫 (𝑅1𝑧))
6665eleq2d 2830 . . . . . . 7 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1𝑧)))
67 elpwi 4325 . . . . . . 7 (𝑦 ∈ 𝒫 (𝑅1𝑧) → 𝑦 ⊆ (𝑅1𝑧))
6866, 67syl6bi 244 . . . . . 6 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ⊆ (𝑅1𝑧)))
69 ssnum 9113 . . . . . 6 (((𝑅1𝑧) ∈ dom card ∧ 𝑦 ⊆ (𝑅1𝑧)) → 𝑦 ∈ dom card)
7063, 68, 69syl6an 674 . . . . 5 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
7170rexlimdva 3178 . . . 4 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧) → 𝑦 ∈ dom card))
721, 71syl5bi 233 . . 3 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑦 (𝑅1 “ On) → 𝑦 ∈ dom card))
7372ssrdv 3767 . 2 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑅1 “ On) ⊆ dom card)
74 onwf 8908 . . . . . 6 On ⊆ (𝑅1 “ On)
7574sseli 3757 . . . . 5 (𝑥 ∈ On → 𝑥 (𝑅1 “ On))
76 pwwf 8885 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
7775, 76sylib 209 . . . 4 (𝑥 ∈ On → 𝒫 𝑥 (𝑅1 “ On))
78 ssel 3755 . . . 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 3112 . 2 ( (𝑅1 “ On) ⊆ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
8173, 80impbii 200 1 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ (𝑅1 “ On) ⊆ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  wss 3732  ifcif 4243  𝒫 cpw 4315   cuni 4594   class class class wbr 4809  cmpt 4888   E cep 5189  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  ccom 5281  Oncon0 5908  suc csuc 5910  1-1wf1 6065  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  recscrecs 7671   +𝑜 coa 7761   ·𝑜 comu 7762  cen 8157  OrdIsocoi 8621  harchar 8668  𝑅1cr1 8840  rankcrnk 8841  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-omul 7769  df-er 7947  df-en 8161  df-dom 8162  df-oi 8622  df-har 8670  df-r1 8842  df-rank 8843  df-card 9016
This theorem is referenced by:  dfac12a  9223
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