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Theorem r1sucg 8617
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 7504 . . 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 8612 . . . 4 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
32dmeqi 5314 . . 3 dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
41, 3eleq2s 2717 . 2 (𝐴 ∈ dom 𝑅1 → (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)))
52fveq1i 6179 . 2 (𝑅1‘suc 𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴)
6 fvex 6188 . . . 4 (𝑅1𝐴) ∈ V
7 pweq 4152 . . . . 5 (𝑥 = (𝑅1𝐴) → 𝒫 𝑥 = 𝒫 (𝑅1𝐴))
8 eqid 2620 . . . . 5 (𝑥 ∈ V ↦ 𝒫 𝑥) = (𝑥 ∈ V ↦ 𝒫 𝑥)
96pwex 4839 . . . . 5 𝒫 (𝑅1𝐴) ∈ V
107, 8, 9fvmpt 6269 . . . 4 ((𝑅1𝐴) ∈ V → ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴))
116, 10ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴)
122fveq1i 6179 . . . 4 (𝑅1𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)
1312fveq2i 6181 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
1411, 13eqtr3i 2644 . 2 𝒫 (𝑅1𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
154, 5, 143eqtr4g 2679 1 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  𝒫 cpw 4149  cmpt 4720  dom cdm 5104  suc csuc 5713  cfv 5876  reccrdg 7490  𝑅1cr1 8610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-r1 8612
This theorem is referenced by:  r1suc  8618  r1fin  8621  r1tr  8624  r1ordg  8626  r1pwss  8632  r1val1  8634  rankwflemb  8641  r1elwf  8644  rankr1ai  8646  rankr1bg  8651  pwwf  8655  unwf  8658  uniwf  8667  rankonidlem  8676  rankr1id  8710
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