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Theorem r1pwcl 9264
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 9213 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
2 elfvdm 6684 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
31, 2jca 515 . . 3 (𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1))
43a1i 11 . 2 (Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)))
5 r1elwf 9213 . . . . 5 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 (𝑅1 “ On))
6 pwwf 9224 . . . . 5 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
75, 6sylibr 237 . . . 4 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
8 elfvdm 6684 . . . 4 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
97, 8jca 515 . . 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 7549 . . . . . 6 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1211adantr 484 . . . . 5 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13 rankpwi 9240 . . . . . . 7 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1413ad2antrl 727 . . . . . 6 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1514eleq1d 2898 . . . . 5 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1612, 15bitr4d 285 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
17 rankr1ag 9219 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1817adantl 485 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19 rankr1ag 9219 . . . . . 6 ((𝒫 𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
206, 19sylanb 584 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantl 485 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2216, 18, 213bitr4d 314 . . 3 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))
2322ex 416 . 2 (Lim 𝐵 → ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵))))
244, 10, 23pm5.21ndd 384 1 (Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  𝒫 cpw 4511   cuni 4813  dom cdm 5532  cima 5535  Oncon0 6169  Lim wlim 6170  suc csuc 6171  cfv 6334  𝑅1cr1 9179  rankcrnk 9180
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181  df-rank 9182
This theorem is referenced by:  r1limwun  10147
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