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Theorem fvmpt2 5600
Description: Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
Hypothesis
Ref Expression
fvmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fvmpt2  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( F `  x
)  =  B )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)    F( x)

Proof of Theorem fvmpt2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3061 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3066 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2eqtrdi 2226 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2319 . . . 4  |-  F/_ y B
6 nfcsb1v 3091 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 3067 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 4099 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2198 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmptg 5593 1  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( F `  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   [_csb 3058    |-> cmpt 4065   ` cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225
This theorem is referenced by:  fvmptssdm  5601  fvmpt2d  5603  fvmptdf  5604  mpteqb  5607  fvmptt  5608  fvmptf  5609  fnmptfvd  5621  ralrnmpt  5659  rexrnmpt  5660  fmptco  5683  f1mpt  5772  offval2  6098  ofrfval2  6099  mptelixpg  6734  dom2lem  6772  mapxpen  6848  xpmapenlem  6849  mkvprop  7156  cc2lem  7265  cc3  7267  fsum3cvg  11386  summodclem2a  11389  fsumf1o  11398  fsum3cvg2  11402  fsumadd  11414  isummulc2  11434  fproddccvg  11580  fprodf1o  11596  txcnp  13774  cnmpt11  13786  cnmpt1t  13788  lgseisenlem2  14454
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