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

Theorem csbfv 6374
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 6373 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴 / 𝑥𝑥))
2 csbvarg 4147 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
32fveq2d 6336 . . 3 (𝐴 ∈ V → (𝐹𝐴 / 𝑥𝑥) = (𝐹𝐴))
41, 3eqtrd 2805 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
5 csbprc 4124 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = ∅)
6 fvprc 6326 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
75, 6eqtr4d 2808 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
84, 7pm2.61i 176 1 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  Vcvv 3351  csb 3682  c0 4063  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-dm 5259  df-iota 5994  df-fv 6039
This theorem is referenced by:  mptcoe1fsupp  19800  mptcoe1matfsupp  20827  mp2pm2mplem4  20834  chfacfscmulfsupp  20884  chfacfpmmulfsupp  20888  cpmidpmatlem3  20897  cayhamlem4  20913  cayleyhamilton1  20917  logbmpt  24747  nbgrcl  26450  nbgrclOLD  26451  nbgrnvtx0  26455  iuninc  29717  disjxpin  29739  cnfinltrel  33578  finixpnum  33727  cdlemkid3N  36742  cdlemkid4  36743  cdlemk39s  36748  mccllem  40347
  Copyright terms: Public domain W3C validator