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Theorem feqmptd 5442
Description: Deduction form of dffn5im 5435. (Contributed by Mario Carneiro, 8-Jan-2015.)
Hypothesis
Ref Expression
feqmptd.1  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
feqmptd  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
Distinct variable groups:    x, A    x, F
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem feqmptd
StepHypRef Expression
1 feqmptd.1 . . 3  |-  ( ph  ->  F : A --> B )
2 ffn 5242 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 14 . 2  |-  ( ph  ->  F  Fn  A )
4 dffn5im 5435 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
53, 4syl 14 1  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    |-> cmpt 3959    Fn wfn 5088   -->wf 5089   ` cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  feqresmpt  5443  cofmpt  5557  fcoconst  5559  suppssof1  5967  ofco  5968  caofinvl  5972  caofcom  5973  mapxpen  6710  xpmapenlem  6711  cnrecnv  10650  lmcn2  12376  cnmpt11f  12380  cnmpt21f  12388  cncfmpt1f  12680  negfcncf  12685  cnrehmeocntop  12689  dvcnp2cntop  12759  dvimulf  12766  dvcoapbr  12767  dvcj  12769  dvfre  12770  dvef  12783
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