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Theorem r1pwcl 8593
 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 8542 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
2 elfvdm 6130 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
31, 2jca 553 . . 3 (𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1))
43a1i 11 . 2 (Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) → (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)))
5 r1elwf 8542 . . . . 5 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 (𝑅1 “ On))
6 pwwf 8553 . . . . 5 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
75, 6sylibr 223 . . . 4 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
8 elfvdm 6130 . . . 4 (𝒫 𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
97, 8jca 553 . . 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 6941 . . . . . 6 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1211adantr 480 . . . . 5 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13 rankpwi 8569 . . . . . . 7 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1413ad2antrl 760 . . . . . 6 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1514eleq1d 2672 . . . . 5 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1612, 15bitr4d 270 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
17 rankr1ag 8548 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1817adantl 481 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19 rankr1ag 8548 . . . . . 6 ((𝒫 𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
206, 19sylanb 488 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantl 481 . . . 4 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝒫 𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2216, 18, 213bitr4d 299 . . 3 ((Lim 𝐵 ∧ (𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))
2322ex 449 . 2 (Lim 𝐵 → ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵))))
244, 10, 23pm5.21ndd 368 1 (Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038   “ cima 5041  Oncon0 5640  Lim wlim 5641  suc csuc 5642  ‘cfv 5804  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511 This theorem is referenced by:  r1limwun  9437
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