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Theorem r1sucg 8880
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 7756 . . 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 8875 . . . 4 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
32dmeqi 5526 . . 3 dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
41, 3eleq2s 2894 . 2 (𝐴 ∈ dom 𝑅1 → (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)))
52fveq1i 6410 . 2 (𝑅1‘suc 𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴)
6 fvex 6422 . . . 4 (𝑅1𝐴) ∈ V
7 pweq 4350 . . . . 5 (𝑥 = (𝑅1𝐴) → 𝒫 𝑥 = 𝒫 (𝑅1𝐴))
8 eqid 2797 . . . . 5 (𝑥 ∈ V ↦ 𝒫 𝑥) = (𝑥 ∈ V ↦ 𝒫 𝑥)
96pwex 5048 . . . . 5 𝒫 (𝑅1𝐴) ∈ V
107, 8, 9fvmpt 6505 . . . 4 ((𝑅1𝐴) ∈ V → ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴))
116, 10ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴)
122fveq1i 6410 . . . 4 (𝑅1𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)
1312fveq2i 6412 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
1411, 13eqtr3i 2821 . 2 𝒫 (𝑅1𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
154, 5, 143eqtr4g 2856 1 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3383  c0 4113  𝒫 cpw 4347  cmpt 4920  dom cdm 5310  suc csuc 5941  cfv 6099  reccrdg 7742  𝑅1cr1 8873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-r1 8875
This theorem is referenced by:  r1suc  8881  r1fin  8884  r1tr  8887  r1ordg  8889  r1pwss  8895  r1val1  8897  rankwflemb  8904  r1elwf  8907  rankr1ai  8909  rankr1bg  8914  pwwf  8918  unwf  8921  uniwf  8930  rankonidlem  8939  rankr1id  8973
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