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Theorem fvmpt2 5577
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 3052 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3057 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2eqtrdi 2219 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2312 . . . 4  |-  F/_ y B
6 nfcsb1v 3082 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 3058 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 4082 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2191 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmptg 5570 1  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( F `  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   [_csb 3049    |-> cmpt 4048   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204
This theorem is referenced by:  fvmptssdm  5578  fvmpt2d  5580  fvmptdf  5581  mpteqb  5584  fvmptt  5585  fvmptf  5586  ralrnmpt  5635  rexrnmpt  5636  fmptco  5659  f1mpt  5747  offval2  6073  ofrfval2  6074  mptelixpg  6708  dom2lem  6746  mapxpen  6822  xpmapenlem  6823  mkvprop  7130  cc2lem  7215  cc3  7217  fsum3cvg  11328  summodclem2a  11331  fsumf1o  11340  fsum3cvg2  11344  fsumadd  11356  isummulc2  11376  fproddccvg  11522  fprodf1o  11538  txcnp  12986  cnmpt11  12998  cnmpt1t  13000
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