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Theorem fvmpt 5344
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 5343 . 2  |-  ( ( A  e.  D  /\  C  e.  _V )  ->  ( F `  A
)  =  C )
51, 4mpan2 416 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   _Vcvv 2615    |-> cmpt 3874   ` cfv 4981
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-iota 4946  df-fun 4983  df-fv 4989
This theorem is referenced by:  reldm  5913  rdg0  6106  oacl  6175  fvmptmap  6394  xpcomco  6494  uzval  8953  sqrtrval  10329  qnumval  11045  qdenval  11046  peano4nninf  11341  peano3nninf  11342  nninfsellemeq  11351
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