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Theorem pwwf 8948
 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 8945 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2 sspwb 5139 . . . . . . 7 (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
31, 2sylib 210 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
4 rankdmr1 8942 . . . . . . 7 (rank‘𝐴) ∈ dom 𝑅1
5 r1sucg 8910 . . . . . . 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 3878 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
8 fvex 6447 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) ∈ V
98elpw2 5051 . . . . 5 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
107, 9sylibr 226 . . . 4 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
11 r1funlim 8907 . . . . . . . 8 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1211simpri 481 . . . . . . 7 Lim dom 𝑅1
13 limsuc 7311 . . . . . . 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 222 . . . . 5 suc (rank‘𝐴) ∈ dom 𝑅1
16 r1sucg 8910 . . . . 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 2918 . . 3 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
19 r1elwf 8937 . . 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 8946 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
22 pwexr 7235 . . . 4 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ V)
23 pwidg 4394 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2422, 23syl 17 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
2521, 24sseldd 3829 . 2 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2620, 25impbii 201 1 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1658   ∈ wcel 2166  Vcvv 3415   ⊆ wss 3799  𝒫 cpw 4379  ∪ cuni 4659  dom cdm 5343   “ cima 5346  Oncon0 5964  Lim wlim 5965  suc csuc 5966  Fun wfun 6118  ‘cfv 6124  𝑅1cr1 8903  rankcrnk 8904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-om 7328  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-r1 8905  df-rank 8906 This theorem is referenced by:  snwf  8950  uniwf  8960  rankpwi  8964  r1pw  8986  r1pwcl  8988  dfac12r  9284  wfgru  9954
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