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Theorem fvmpts 5807
Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fvmpts.1  |-  F  =  ( x  e.  C  |->  B )
Assertion
Ref Expression
fvmpts  |-  ( ( A  e.  C  /\  [_ A  /  x ]_ B  e.  V )  ->  ( F `  A
)  =  [_ A  /  x ]_ B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    F( x)    V( x)

Proof of Theorem fvmpts
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3254 . 2  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
2 fvmpts.1 . . 3  |-  F  =  ( x  e.  C  |->  B )
3 nfcv 2572 . . . 4  |-  F/_ y B
4 nfcsb1v 3283 . . . 4  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3259 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4299 . . 3  |-  ( x  e.  C  |->  B )  =  ( y  e.  C  |->  [_ y  /  x ]_ B )
72, 6eqtri 2456 . 2  |-  F  =  ( y  e.  C  |-> 
[_ y  /  x ]_ B )
81, 7fvmptg 5804 1  |-  ( ( A  e.  C  /\  [_ A  /  x ]_ B  e.  V )  ->  ( F `  A
)  =  [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   [_csb 3251    e. cmpt 4266   ` cfv 5454
This theorem is referenced by:  fvmptd  5810  pcmpt  13261  issubc  14035  prdsdsf  18397  itgparts  19931  dchrisumlema  21182  abfmpeld  24066  abfmpel  24067  prodss  25273  fprodser  25275  fprodefsum  25298  fprodn0  25303  aomclem6  27134  cdlemk40  31714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462
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