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Theorem r1sucg 8488
Description: Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1sucg (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))

Proof of Theorem r1sucg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rdgsucg 7379 . . 3 (𝐴 ∈ dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) → (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)))
2 df-r1 8483 . . . 4 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
32dmeqi 5230 . . 3 dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
41, 3eleq2s 2701 . 2 (𝐴 ∈ dom 𝑅1 → (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)))
52fveq1i 6085 . 2 (𝑅1‘suc 𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴)
6 fvex 6094 . . . 4 (𝑅1𝐴) ∈ V
7 pweq 4106 . . . . 5 (𝑥 = (𝑅1𝐴) → 𝒫 𝑥 = 𝒫 (𝑅1𝐴))
8 eqid 2605 . . . . 5 (𝑥 ∈ V ↦ 𝒫 𝑥) = (𝑥 ∈ V ↦ 𝒫 𝑥)
96pwex 4765 . . . . 5 𝒫 (𝑅1𝐴) ∈ V
107, 8, 9fvmpt 6172 . . . 4 ((𝑅1𝐴) ∈ V → ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴))
116, 10ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴)
122fveq1i 6085 . . . 4 (𝑅1𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)
1312fveq2i 6087 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
1411, 13eqtr3i 2629 . 2 𝒫 (𝑅1𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
154, 5, 143eqtr4g 2664 1 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  Vcvv 3168  c0 3869  𝒫 cpw 4103  cmpt 4633  dom cdm 5024  suc csuc 5624  cfv 5786  reccrdg 7365  𝑅1cr1 8481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-r1 8483
This theorem is referenced by:  r1suc  8489  r1fin  8492  r1tr  8495  r1ordg  8497  r1pwss  8503  r1val1  8505  rankwflemb  8512  r1elwf  8515  rankr1ai  8517  rankr1bg  8522  pwwf  8526  unwf  8529  uniwf  8538  rankonidlem  8547  rankr1id  8581
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