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

Theorem csbfv 6942
Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
Distinct variable group:   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem csbfv
StepHypRef Expression
1 csbfv2g 6941 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴 / 𝑥𝑥))
2 csbvarg 4432 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
32fveq2d 6896 . . 3 (𝐴 ∈ V → (𝐹𝐴 / 𝑥𝑥) = (𝐹𝐴))
41, 3eqtrd 2773 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
5 csbprc 4407 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = ∅)
6 fvprc 6884 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
75, 6eqtr4d 2776 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
84, 7pm2.61i 182 1 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  csb 3894  c0 4323  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-dm 5687  df-iota 6496  df-fv 6552
This theorem is referenced by:  mptcoe1fsupp  21739  mptcoe1matfsupp  22304  mp2pm2mplem4  22311  chfacfscmulfsupp  22361  chfacfpmmulfsupp  22365  cpmidpmatlem3  22374  cayhamlem4  22390  cayleyhamilton1  22394  logbmpt  26293  nbgrcl  28592  nbgrnvtx0  28596  iuninc  31792  disjxpin  31819  finixpnum  36473  cdlemkid3N  39804  cdlemkid4  39805  cdlemk39s  39810  mccllem  44313
  Copyright terms: Public domain W3C validator