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Theorem pwwf 8621
 Description: A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
pwwf (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))

Proof of Theorem pwwf
StepHypRef Expression
1 r1rankidb 8618 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2 sspwb 4883 . . . . . . 7 (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
31, 2sylib 208 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
4 rankdmr1 8615 . . . . . . 7 (rank‘𝐴) ∈ dom 𝑅1
5 r1sucg 8583 . . . . . . 7 ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
64, 5ax-mp 5 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
73, 6syl6sseqr 3636 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
8 fvex 6163 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) ∈ V
98elpw2 4793 . . . . 5 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
107, 9sylibr 224 . . . 4 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
11 r1funlim 8580 . . . . . . . 8 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1211simpri 478 . . . . . . 7 Lim dom 𝑅1
13 limsuc 7003 . . . . . . 7 (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1))
1412, 13ax-mp 5 . . . . . 6 ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)
154, 14mpbi 220 . . . . 5 suc (rank‘𝐴) ∈ dom 𝑅1
16 r1sucg 8583 . . . . 5 (suc (rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
1715, 16ax-mp 5 . . . 4 (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
1810, 17syl6eleqr 2709 . . 3 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
19 r1elwf 8610 . . 3 (𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)) → 𝒫 𝐴 (𝑅1 “ On))
2018, 19syl 17 . 2 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
21 r1elssi 8619 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
22 elex 3201 . . . . 5 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ V)
23 pwexb 6929 . . . . 5 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2422, 23sylibr 224 . . . 4 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ V)
25 pwidg 4149 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2624, 25syl 17 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
2721, 26sseldd 3588 . 2 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2820, 27impbii 199 1 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1480   ∈ wcel 1987  Vcvv 3189   ⊆ wss 3559  𝒫 cpw 4135  ∪ cuni 4407  dom cdm 5079   “ cima 5082  Oncon0 5687  Lim wlim 5688  suc csuc 5689  Fun wfun 5846  ‘cfv 5852  𝑅1cr1 8576  rankcrnk 8577 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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  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 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8578  df-rank 8579 This theorem is referenced by:  snwf  8623  uniwf  8633  rankpwi  8637  r1pw  8659  r1pwcl  8661  dfac12r  8919  wfgru  9589
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