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Theorem fvmpt 5277
Description: Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
Hypotheses
Ref Expression
fvmptg.1  |-  ( x  =  A  ->  B  =  C )
fvmptg.2  |-  F  =  ( x  e.  D  |->  B )
fvmpt.3  |-  C  e. 
_V
Assertion
Ref Expression
fvmpt  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmpt
StepHypRef Expression
1 fvmpt.3 . 2  |-  C  e. 
_V
2 fvmptg.1 . . 3  |-  ( x  =  A  ->  B  =  C )
3 fvmptg.2 . . 3  |-  F  =  ( x  e.  D  |->  B )
42, 3fvmptg 5276 . 2  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  C )
51, 4mpan2 409 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409   _Vcvv 2574    |-> 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-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:  reldm  5840  rdg0  6005  oacl  6071  xpcomco  6331  uzval  8571  sqrtrval  9827
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