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Theorem fvmpts 5499
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 3006 . 2  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
2 fvmpts.1 . . 3  |-  F  =  ( x  e.  C  |->  B )
3 nfcv 2281 . . . 4  |-  F/_ y B
4 nfcsb1v 3035 . . . 4  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3012 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4023 . . 3  |-  ( x  e.  C  |->  B )  =  ( y  e.  C  |->  [_ y  /  x ]_ B )
72, 6eqtri 2160 . 2  |-  F  =  ( y  e.  C  |-> 
[_ y  /  x ]_ B )
81, 7fvmptg 5497 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 103    = wceq 1331    e. wcel 1480   [_csb 3003    |-> cmpt 3989   ` cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131
This theorem is referenced by:  fvmptd  5502  cc3  7090  sumfct  11157  zsumdc  11167  isumss  11174  fsummulc2  11231  isumshft  11273  mulcncflem  12775
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