MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1sucg Structured version   Visualization version   GIF version

Theorem r1sucg 9809
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 8463 . . 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 9804 . . . 4 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
32dmeqi 5915 . . 3 dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
41, 3eleq2s 2859 . 2 (𝐴 ∈ dom 𝑅1 → (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)))
52fveq1i 6907 . 2 (𝑅1‘suc 𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴)
6 fvex 6919 . . . 4 (𝑅1𝐴) ∈ V
7 pweq 4614 . . . . 5 (𝑥 = (𝑅1𝐴) → 𝒫 𝑥 = 𝒫 (𝑅1𝐴))
8 eqid 2737 . . . . 5 (𝑥 ∈ V ↦ 𝒫 𝑥) = (𝑥 ∈ V ↦ 𝒫 𝑥)
96pwex 5380 . . . . 5 𝒫 (𝑅1𝐴) ∈ V
107, 8, 9fvmpt 7016 . . . 4 ((𝑅1𝐴) ∈ V → ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴))
116, 10ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = 𝒫 (𝑅1𝐴)
122fveq1i 6907 . . . 4 (𝑅1𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)
1312fveq2i 6909 . . 3 ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1𝐴)) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
1411, 13eqtr3i 2767 . 2 𝒫 (𝑅1𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))
154, 5, 143eqtr4g 2802 1 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  𝒫 cpw 4600  cmpt 5225  dom cdm 5685  suc csuc 6386  cfv 6561  reccrdg 8449  𝑅1cr1 9802
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804
This theorem is referenced by:  r1suc  9810  r1fin  9813  r1tr  9816  r1ordg  9818  r1pwss  9824  r1val1  9826  rankwflemb  9833  r1elwf  9836  rankr1ai  9838  rankr1bg  9843  pwwf  9847  unwf  9850  uniwf  9859  rankonidlem  9868  rankr1id  9902
  Copyright terms: Public domain W3C validator