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Theorem fvco4 5500
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 5499 . . 3  |-  ( ( K : A --> X  /\  u  e.  A )  ->  ( ( H  o.  K ) `  u
)  =  ( H `
 ( K `  u ) ) )
21ad2ant2r 501 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( ( H  o.  K ) `  u )  =  ( H `  ( K `
 u ) ) )
3 simplr 520 . . . 4  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H  o.  K )  =  F )
43eqcomd 2146 . . 3  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  F  =  ( H  o.  K ) )
54fveq1d 5430 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( F `  u )  =  ( ( H  o.  K
) `  u )
)
6 fveq2 5428 . . 3  |-  ( x  =  ( K `  u )  ->  ( H `  x )  =  ( H `  ( K `  u ) ) )
76ad2antll 483 . 2  |-  ( ( ( K : A --> X  /\  ( H  o.  K )  =  F )  /\  ( u  e.  A  /\  x  =  ( K `  u ) ) )  ->  ( H `  x )  =  ( H `  ( K `
 u ) ) )
82, 5, 73eqtr4rd 2184 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 1332    e. wcel 1481    o. ccom 4550   -->wf 5126   ` cfv 5130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138
This theorem is referenced by: (None)
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