MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac12r Structured version   Visualization version   GIF version

Theorem dfac12r 8913
Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 8916 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 8601 . . . 4 (𝑦 (𝑅1 “ On) ↔ ∃𝑧 ∈ On 𝑦 ∈ (𝑅1‘suc 𝑧))
2 harcl 8411 . . . . . . . . 9 (har‘(𝑅1𝑧)) ∈ On
3 pweq 4138 . . . . . . . . . . 11 (𝑥 = (har‘(𝑅1𝑧)) → 𝒫 𝑥 = 𝒫 (har‘(𝑅1𝑧)))
43eleq1d 2688 . . . . . . . . . 10 (𝑥 = (har‘(𝑅1𝑧)) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (har‘(𝑅1𝑧)) ∈ dom card))
54rspcv 3296 . . . . . . . . 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 8724 . . . . . . . 8 (𝒫 (har‘(𝑅1𝑧)) ∈ dom card → (card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)))
8 ensym 7950 . . . . . . . 8 ((card‘𝒫 (har‘(𝑅1𝑧))) ≈ 𝒫 (har‘(𝑅1𝑧)) → 𝒫 (har‘(𝑅1𝑧)) ≈ (card‘𝒫 (har‘(𝑅1𝑧))))
9 bren 7909 . . . . . . . . 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 6095 . . . . . . . . . . . . . 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 8715 . . . . . . . . . . . . . 14 (card‘𝒫 (har‘(𝑅1𝑧))) ∈ On
1413onssi 6985 . . . . . . . . . . . . 13 (card‘𝒫 (har‘(𝑅1𝑧))) ⊆ On
15 f1ss 6065 . . . . . . . . . . . . 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 693 . . . . . . . . . . . 12 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → 𝑓:𝒫 (har‘(𝑅1𝑧))–1-1→On)
17 fveq2 6150 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (rank‘𝑦) = (rank‘𝑏))
1817oveq2d 6621 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) = (suc ran ran 𝑥 ·𝑜 (rank‘𝑏)))
19 suceq 5752 . . . . . . . . . . . . . . . . . . . . 21 ((rank‘𝑦) = (rank‘𝑏) → suc (rank‘𝑦) = suc (rank‘𝑏))
2017, 19syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → suc (rank‘𝑦) = suc (rank‘𝑏))
2120fveq2d 6154 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (𝑥‘suc (rank‘𝑦)) = (𝑥‘suc (rank‘𝑏)))
22 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏𝑦 = 𝑏)
2321, 22fveq12d 6156 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((𝑥‘suc (rank‘𝑦))‘𝑦) = ((𝑥‘suc (rank‘𝑏))‘𝑏))
2418, 23oveq12d 6623 . . . . . . . . . . . . . . . . 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 5425 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏))
2625fveq2d 6154 . . . . . . . . . . . . . . . . 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 4083 . . . . . . . . . . . . . . . 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 4715 . . . . . . . . . . . . . . 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 5289 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → dom 𝑥 = dom 𝑎)
3029fveq2d 6154 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑅1‘dom 𝑥) = (𝑅1‘dom 𝑎))
3129unieqd 4417 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 dom 𝑥 = dom 𝑎)
3229, 31eqeq12d 2641 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (dom 𝑥 = dom 𝑥 ↔ dom 𝑎 = dom 𝑎))
33 rneq 5315 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → ran 𝑥 = ran 𝑎)
3433unieqd 4417 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 ran 𝑥 = ran 𝑎)
3534rneqd 5317 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → ran ran 𝑥 = ran ran 𝑎)
3635unieqd 4417 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 ran ran 𝑥 = ran ran 𝑎)
37 suceq 5752 . . . . . . . . . . . . . . . . . . . 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 6620 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (suc ran ran 𝑥 ·𝑜 (rank‘𝑏)) = (suc ran ran 𝑎 ·𝑜 (rank‘𝑏)))
40 fveq1 6149 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥‘suc (rank‘𝑏)) = (𝑎‘suc (rank‘𝑏)))
4140fveq1d 6152 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥‘suc (rank‘𝑏))‘𝑏) = ((𝑎‘suc (rank‘𝑏))‘𝑏))
4239, 41oveq12d 6623 . . . . . . . . . . . . . . . . 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 6156 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑎 → (𝑥 dom 𝑥) = (𝑎 dom 𝑎))
4544rneqd 5317 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → ran (𝑥 dom 𝑥) = ran (𝑎 dom 𝑎))
46 oieq2 8363 . . . . . . . . . . . . . . . . . . . . . 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 5263 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran (𝑎 dom 𝑎)))
4948, 44coeq12d 5251 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)))
5049imaeq1d 5428 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑏) = ((OrdIso( E , ran (𝑎 dom 𝑎)) ∘ (𝑎 dom 𝑎)) “ 𝑏))
5150fveq2d 6154 . . . . . . . . . . . . . . . . 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 4090 . . . . . . . . . . . . . . . 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 4698 . . . . . . . . . . . . . . 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 2672 . . . . . . . . . . . . . 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 4715 . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . 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 8912 . . . . . . . . . . 11 ((𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
5958ex 450 . . . . . . . . . 10 (𝑓:𝒫 (har‘(𝑅1𝑧))–1-1-onto→(card‘𝒫 (har‘(𝑅1𝑧))) → (𝑧 ∈ On → (𝑅1𝑧) ∈ dom card))
6059exlimiv 1860 . . . . . . . . 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 445 . . . . . 6 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑅1𝑧) ∈ dom card)
64 r1suc 8578 . . . . . . . . 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 2689 . . . . . . 7 ((∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ∧ 𝑧 ∈ On) → (𝑦 ∈ (𝑅1‘suc 𝑧) ↔ 𝑦 ∈ 𝒫 (𝑅1𝑧)))
67 elpwi 4145 . . . . . . 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 8807 . . . . . 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 3029 . . . 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 3594 . 2 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → (𝑅1 “ On) ⊆ dom card)
74 onwf 8638 . . . . . 6 On ⊆ (𝑅1 “ On)
7574sseli 3584 . . . . 5 (𝑥 ∈ On → 𝑥 (𝑅1 “ On))
76 pwwf 8615 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
7775, 76sylib 208 . . . 4 (𝑥 ∈ On → 𝒫 𝑥 (𝑅1 “ On))
78 ssel 3582 . . . 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 2964 . 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 384   = wceq 1480  wex 1701  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  wss 3560  ifcif 4063  𝒫 cpw 4135   cuni 4407   class class class wbr 4618  cmpt 4678   E cep 4988  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  ccom 5083  Oncon0 5685  suc csuc 5687  1-1wf1 5847  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  recscrecs 7413   +𝑜 coa 7503   ·𝑜 comu 7504  cen 7897  OrdIsocoi 8359  harchar 8406  𝑅1cr1 8570  rankcrnk 8571  cardccrd 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510  df-omul 7511  df-er 7688  df-en 7901  df-dom 7902  df-oi 8360  df-har 8408  df-r1 8572  df-rank 8573  df-card 8710
This theorem is referenced by:  dfac12a  8915
  Copyright terms: Public domain W3C validator