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Theorem fvmpts 5603
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 3084 . 2  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
2 fvmpts.1 . . 3  |-  F  =  ( x  e.  C  |->  B )
3 nfcv 2419 . . . 4  |-  F/_ y B
4 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4110 . . 3  |-  ( x  e.  C  |->  B )  =  ( y  e.  C  |->  [_ y  /  x ]_ B )
72, 6eqtri 2303 . 2  |-  F  =  ( y  e.  C  |-> 
[_ y  /  x ]_ B )
81, 7fvmptg 5600 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 358    = wceq 1623    e. wcel 1684   [_csb 3081    e. cmpt 4077   ` cfv 5255
This theorem is referenced by:  fvmptd  5606  pcmpt  12940  issubc  13712  prdsdsf  17931  itgparts  19394  dchrisumlema  20637  abfmpeld  23218  abfmpel  23219  aomclem6  27156  cdlemk40  31106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263
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