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Theorem fvco4 5609
Description: Value of a composition. (Contributed by BJ, 7-Jul-2022.)
Assertion
Ref Expression
fvco4  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H `  x )  =  ( F `  u ) )

Proof of Theorem fvco4
StepHypRef Expression
1 fvco3 5608 . . 3  |-  ( ( K : A --> X  /\  u  e.  A )  ->  ( ( H  o.  K ) `  u
)  =  ( H `
 ( K `  u ) ) )
21ad2ant2r 509 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( ( H  o.  K ) `  u )  =  ( H `  ( K `
 u ) ) )
3 simplr 528 . . . 4  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H  o.  K )  =  F )
43eqcomd 2195 . . 3  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  F  =  ( H  o.  K ) )
54fveq1d 5536 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( F `  u )  =  ( ( H  o.  K
) `  u )
)
6 fveq2 5534 . . 3  |-  ( x  =  ( K `  u )  ->  ( H `  x )  =  ( H `  ( K `  u ) ) )
76ad2antll 491 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H `  x )  =  ( H `  ( K `
 u ) ) )
82, 5, 73eqtr4rd 2233 1  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H `  x )  =  ( F `  u ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    o. ccom 4648   -->wf 5231   ` cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243
This theorem is referenced by: (None)
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