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Theorem pwwf 9784
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 9781 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
21sspwd 4609 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
3 rankdmr1 9778 . . . . . . 7 (rank‘𝐴) ∈ dom 𝑅1
4 r1sucg 9746 . . . . . . 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 4029 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
7 fvex 6891 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) ∈ V
87elpw2 5338 . . . . 5 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
96, 8sylibr 233 . . . 4 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
10 r1funlim 9743 . . . . . . . 8 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1110simpri 486 . . . . . . 7 Lim dom 𝑅1
12 limsuc 7821 . . . . . . 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 229 . . . . 5 suc (rank‘𝐴) ∈ dom 𝑅1
15 r1sucg 9746 . . . . 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 2843 . . 3 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
18 r1elwf 9773 . . 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 9782 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
21 pwexr 7735 . . . 4 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ V)
22 pwidg 4616 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
2321, 22syl 17 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
2420, 23sseldd 3979 . 2 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2519, 24impbii 208 1 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  Vcvv 3473  wss 3944  𝒫 cpw 4596   cuni 4901  dom cdm 5669  cima 5672  Oncon0 6353  Lim wlim 6354  suc csuc 6355  Fun wfun 6526  cfv 6532  𝑅1cr1 9739  rankcrnk 9740
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-r1 9741  df-rank 9742
This theorem is referenced by:  snwf  9786  uniwf  9796  rankpwi  9800  r1pw  9822  r1pwcl  9824  dfac12r  10123  wfgru  10793
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