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Theorem feqresmpt 5569
Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
feqmptd.1  |-  ( ph  ->  F : A --> B )
feqresmpt.2  |-  ( ph  ->  C  C_  A )
Assertion
Ref Expression
feqresmpt  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
Distinct variable groups:    x, A    x, C    x, F
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem feqresmpt
StepHypRef Expression
1 feqmptd.1 . . . 4  |-  ( ph  ->  F : A --> B )
2 feqresmpt.2 . . . 4  |-  ( ph  ->  C  C_  A )
3 fssres 5390 . . . 4  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
41, 2, 3syl2anc 411 . . 3  |-  ( ph  ->  ( F  |`  C ) : C --> B )
54feqmptd 5568 . 2  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( ( F  |`  C ) `  x
) ) )
6 fvres 5538 . . 3  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
76mpteq2ia 4088 . 2  |-  ( x  e.  C  |->  ( ( F  |`  C ) `  x ) )  =  ( x  e.  C  |->  ( F `  x
) )
85, 7eqtrdi 2226 1  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    C_ wss 3129    |-> cmpt 4063    |` cres 4627   -->wf 5211   ` cfv 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223
This theorem is referenced by:  dvmulxxbr  14037
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