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Theorem uniwf 9251
 Description: A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
uniwf (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))

Proof of Theorem uniwf
StepHypRef Expression
1 r1tr 9208 . . . . . . . 8 Tr (𝑅1‘suc (rank‘𝐴))
2 rankidb 9232 . . . . . . . 8 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
3 trss 5184 . . . . . . . 8 (Tr (𝑅1‘suc (rank‘𝐴)) → (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴))))
41, 2, 3mpsyl 68 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
5 rankdmr1 9233 . . . . . . . 8 (rank‘𝐴) ∈ dom 𝑅1
6 r1sucg 9201 . . . . . . . 8 ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
75, 6ax-mp 5 . . . . . . 7 (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
84, 7sseqtrdi 4020 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
9 sspwuni 5025 . . . . . 6 (𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
108, 9sylib 220 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
11 fvex 6686 . . . . . 6 (𝑅1‘(rank‘𝐴)) ∈ V
1211elpw2 5251 . . . . 5 ( 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1310, 12sylibr 236 . . . 4 (𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
1413, 7eleqtrrdi 2927 . . 3 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
15 r1elwf 9228 . . 3 ( 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
1614, 15syl 17 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
17 pwwf 9239 . . 3 ( 𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
18 pwuni 4878 . . . 4 𝐴 ⊆ 𝒫 𝐴
19 sswf 9240 . . . 4 ((𝒫 𝐴 (𝑅1 “ On) ∧ 𝐴 ⊆ 𝒫 𝐴) → 𝐴 (𝑅1 “ On))
2018, 19mpan2 689 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2117, 20sylbi 219 . 2 ( 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2216, 21impbii 211 1 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1536   ∈ wcel 2113   ⊆ wss 3939  𝒫 cpw 4542  ∪ cuni 4841  Tr wtr 5175  dom cdm 5558   “ cima 5561  Oncon0 6194  suc csuc 6196  ‘cfv 6358  𝑅1cr1 9194  rankcrnk 9195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-r1 9196  df-rank 9197 This theorem is referenced by:  rankuni2b  9285  r1limwun  10161  wfgru  10241
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