Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csbrecsg Structured version   Visualization version   GIF version

Theorem csbrecsg 33629
Description: Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbrecsg (𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))

Proof of Theorem csbrecsg
StepHypRef Expression
1 csbwrecsg 33628 . . 3 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
2 csbconstg 3706 . . . 4 (𝐴𝑉𝐴 / 𝑥 E = E )
3 wrecseq1 7617 . . . 4 (𝐴 / 𝑥 E = E → wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
42, 3syl 17 . . 3 (𝐴𝑉 → wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
5 csbconstg 3706 . . . 4 (𝐴𝑉𝐴 / 𝑥On = On)
6 wrecseq2 7618 . . . 4 (𝐴 / 𝑥On = On → wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
75, 6syl 17 . . 3 (𝐴𝑉 → wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
81, 4, 73eqtrd 2803 . 2 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
9 df-recs 7676 . . 3 recs(𝐹) = wrecs( E , On, 𝐹)
109csbeq2i 4156 . 2 𝐴 / 𝑥recs(𝐹) = 𝐴 / 𝑥wrecs( E , On, 𝐹)
11 df-recs 7676 . 2 recs(𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹)
128, 10, 113eqtr4g 2824 1 (𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  csb 3693   E cep 5191  Oncon0 5910  wrecscwrecs 7613  recscrecs 7675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-xp 5285  df-cnv 5287  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-iota 6033  df-fv 6078  df-wrecs 7614  df-recs 7676
This theorem is referenced by:  csbrdgg  33630
  Copyright terms: Public domain W3C validator