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Theorem fvco4 5425
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 5424 . . 3  |-  ( ( K : A --> X  /\  u  e.  A )  ->  ( ( H  o.  K ) `  u
)  =  ( H `
 ( K `  u ) ) )
21ad2ant2r 496 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( ( H  o.  K ) `  u )  =  ( H `  ( K `
 u ) ) )
3 simplr 500 . . . 4  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H  o.  K )  =  F )
43eqcomd 2105 . . 3  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  F  =  ( H  o.  K ) )
54fveq1d 5355 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( F `  u )  =  ( ( H  o.  K
) `  u )
)
6 fveq2 5353 . . 3  |-  ( x  =  ( K `  u )  ->  ( H `  x )  =  ( H `  ( K `  u ) ) )
76ad2antll 478 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H `  x )  =  ( H `  ( K `
 u ) ) )
82, 5, 73eqtr4rd 2143 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 103    = wceq 1299    e. wcel 1448    o. ccom 4481   -->wf 5055   ` cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067
This theorem is referenced by: (None)
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