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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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