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Theorem fvmpt2 5282
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 2883 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 2887 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2104 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2194 . . . 4  |-  F/_ y B
6 nfcsb1v 2910 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 2888 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 3879 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2076 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmptg 5276 1  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( F `  x
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   [_csb 2880    |-> cmpt 3846   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938
This theorem is referenced by:  fvmptssdm  5283  fvmpt2d  5285  fvmptdf  5286  mpteqb  5289  fvmptt  5290  fvmptf  5291  ralrnmpt  5337  rexrnmpt  5338  fmptco  5358  f1mpt  5438  offval2  5754  ofrfval2  5755  dom2lem  6283
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