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Theorem fvmpt2 5599
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 3060 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3065 . . 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 3090 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 3066 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 4098 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2198 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmptg 5592 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 3057    |-> cmpt 4064   ` cfv 5216
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 4121  ax-pow 4174  ax-pr 4209
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 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224
This theorem is referenced by:  fvmptssdm  5600  fvmpt2d  5602  fvmptdf  5603  mpteqb  5606  fvmptt  5607  fvmptf  5608  fnmptfvd  5620  ralrnmpt  5658  rexrnmpt  5659  fmptco  5682  f1mpt  5771  offval2  6097  ofrfval2  6098  mptelixpg  6733  dom2lem  6771  mapxpen  6847  xpmapenlem  6848  mkvprop  7155  cc2lem  7264  cc3  7266  fsum3cvg  11385  summodclem2a  11388  fsumf1o  11397  fsum3cvg2  11401  fsumadd  11413  isummulc2  11433  fproddccvg  11579  fprodf1o  11595  txcnp  13741  cnmpt11  13753  cnmpt1t  13755  lgseisenlem2  14421
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