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Theorem cofmpt 5685
Description: Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
Hypotheses
Ref Expression
cofmpt.1  |-  ( ph  ->  F : C --> D )
cofmpt.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
Assertion
Ref Expression
cofmpt  |-  ( ph  ->  ( F  o.  (
x  e.  A  |->  B ) )  =  ( x  e.  A  |->  ( F `  B ) ) )
Distinct variable groups:    x, A    x, C    x, F    ph, x
Allowed substitution hints:    B( x)    D( x)

Proof of Theorem cofmpt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cofmpt.2 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
2 eqidd 2178 . 2  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B ) )
3 cofmpt.1 . . 3  |-  ( ph  ->  F : C --> D )
43feqmptd 5569 . 2  |-  ( ph  ->  F  =  ( y  e.  C  |->  ( F `
 y ) ) )
5 fveq2 5515 . 2  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
61, 2, 4, 5fmptco 5682 1  |-  ( ph  ->  ( F  o.  (
x  e.  A  |->  B ) )  =  ( x  e.  A  |->  ( F `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    |-> cmpt 4064    o. ccom 4630   -->wf 5212   ` cfv 5216
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 4121  ax-pow 4174  ax-pr 4209
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-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  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 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224
This theorem is referenced by:  dvcjbr  14103  dvmptcjx  14117  dvef  14119
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