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Theorem fvmpt2 5386
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 2936 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 2940 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2136 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2228 . . . 4  |-  F/_ y B
6 nfcsb1v 2963 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 2941 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 3933 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2108 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmptg 5380 1  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( F `  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   [_csb 2933    |-> cmpt 3899   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023
This theorem is referenced by:  fvmptssdm  5387  fvmpt2d  5389  fvmptdf  5390  mpteqb  5393  fvmptt  5394  fvmptf  5395  ralrnmpt  5441  rexrnmpt  5442  fmptco  5464  f1mpt  5550  offval2  5870  ofrfval2  5871  dom2lem  6487  mapxpen  6562  xpmapenlem  6563  fisumcvg  10762  fsum3cvg  10763  isummolem2a  10767  fsumf1o  10778  fisumcvg2  10782  fsum3cvg2  10783  fsumadd  10796  isummulc2  10816
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