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Theorem pwwf 9269
 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 9266 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
21sspwd 4509 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
3 rankdmr1 9263 . . . . . . 7 (rank‘𝐴) ∈ dom 𝑅1
4 r1sucg 9231 . . . . . . 7 ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
53, 4ax-mp 5 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
62, 5sseqtrrdi 3943 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
7 fvex 6671 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) ∈ V
87elpw2 5215 . . . . 5 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
96, 8sylibr 237 . . . 4 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
10 r1funlim 9228 . . . . . . . 8 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1110simpri 489 . . . . . . 7 Lim dom 𝑅1
12 limsuc 7563 . . . . . . 7 (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1))
1311, 12ax-mp 5 . . . . . 6 ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)
143, 13mpbi 233 . . . . 5 suc (rank‘𝐴) ∈ dom 𝑅1
15 r1sucg 9231 . . . . 5 (suc (rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
1614, 15ax-mp 5 . . . 4 (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
179, 16eleqtrrdi 2863 . . 3 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
18 r1elwf 9258 . . 3 (𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)) → 𝒫 𝐴 (𝑅1 “ On))
1917, 18syl 17 . 2 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
20 r1elssi 9267 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
21 pwexr 7486 . . . 4 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ V)
22 pwidg 4516 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2321, 22syl 17 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
2420, 23sseldd 3893 . 2 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2519, 24impbii 212 1 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  𝒫 cpw 4494  ∪ cuni 4798  dom cdm 5524   “ cima 5527  Oncon0 6169  Lim wlim 6170  suc csuc 6171  Fun wfun 6329  ‘cfv 6335  𝑅1cr1 9224  rankcrnk 9225 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-r1 9226  df-rank 9227 This theorem is referenced by:  snwf  9271  uniwf  9281  rankpwi  9285  r1pw  9307  r1pwcl  9309  dfac12r  9606  wfgru  10276
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