ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cofmpt Unicode version

Theorem cofmpt 5687
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 5571 . 2  |-  ( ph  ->  F  =  ( y  e.  C  |->  ( F `
 y ) ) )
5 fveq2 5517 . 2  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
61, 2, 4, 5fmptco 5684 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 4066    o. ccom 4632   -->wf 5214   ` cfv 5218
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 4123  ax-pow 4176  ax-pr 4211
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 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226
This theorem is referenced by:  dvcjbr  14257  dvmptcjx  14271  dvef  14273
  Copyright terms: Public domain W3C validator