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Theorem r1pwcl 8655
Description: The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
r1pwcl (Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))

Proof of Theorem r1pwcl
StepHypRef Expression
1 r1elwf 8604 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
2 elfvdm 6178 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
31, 2jca 554 . . 3 (𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1))
43a1i 11 . 2 (Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)))
5 r1elwf 8604 . . . . 5 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 (𝑅1 “ On))
6 pwwf 8615 . . . . 5 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
75, 6sylibr 224 . . . 4 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
8 elfvdm 6178 . . . 4 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
97, 8jca 554 . . 3 (𝒫 𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1))
109a1i 11 . 2 (Lim 𝐵 → (𝒫 𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)))
11 limsuc 6997 . . . . . 6 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1211adantr 481 . . . . 5 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13 rankpwi 8631 . . . . . . 7 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1413ad2antrl 763 . . . . . 6 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1514eleq1d 2688 . . . . 5 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1612, 15bitr4d 271 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
17 rankr1ag 8610 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1817adantl 482 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19 rankr1ag 8610 . . . . . 6 ((𝒫 𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
206, 19sylanb 489 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantl 482 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2216, 18, 213bitr4d 300 . . 3 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))
2322ex 450 . 2 (Lim 𝐵 → ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵))))
244, 10, 23pm5.21ndd 369 1 (Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  𝒫 cpw 4135   cuni 4407  dom cdm 5079  cima 5082  Oncon0 5685  Lim wlim 5686  suc csuc 5687  cfv 5850  𝑅1cr1 8570  rankcrnk 8571
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-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-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-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-r1 8572  df-rank 8573
This theorem is referenced by:  r1limwun  9503
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