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Theorem csbfv 5909
Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv  |-  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
)
Distinct variable group:    x, F
Allowed substitution hint:    A( x)

Proof of Theorem csbfv
StepHypRef Expression
1 csbfv2g 5908 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  [_ A  /  x ]_ x ) )
2 csbvarg 3855 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ x  =  A )
32fveq2d 5876 . . 3  |-  ( A  e.  _V  ->  ( F `  [_ A  /  x ]_ x )  =  ( F `  A
) )
41, 3eqtrd 2498 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
) )
5 csbprc 3830 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F `  x )  =  (/) )
6 fvprc 5866 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
75, 6eqtr4d 2501 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F `  x )  =  ( F `  A ) )
84, 7pm2.61i 164 1  |-  [_ A  /  x ]_ ( F `
 x )  =  ( F `  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   [_csb 3430   (/)c0 3793   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586  ax-pow 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-dm 5018  df-iota 5557  df-fv 5602
This theorem is referenced by:  mptcoe1fsupp  18382  mptcoe1matfsupp  19430  mp2pm2mplem4  19437  chfacfscmulfsupp  19487  chfacfpmmulfsupp  19491  cpmidpmatlem3  19500  cayhamlem4  19516  cayleyhamilton1  19520  iuninc  27567  disjxpin  27587  finixpnum  30243  mccllem  31808  cdlemkid3N  36802  cdlemkid4  36803  cdlemk39s  36808
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