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

Step	Hyp	Ref	Expression
1		fvco3 5424	. . 3 |- ( ( K : A --> X /\ u e. A ) -> ( ( H o. K ) ` u ) = ( H ` ( K ` u ) ) )
2	1	ad2ant2r 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 )
4	3	eqcomd 2105	. . 3 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> F = ( H o. K ) )
5	4	fveq1d 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 ) ) )
7	6	ad2antll 478	. 2 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( H ` x ) = ( H ` ( K ` u ) ) )
8	2, 5, 7	3eqtr4rd 2143	1 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( H ` x ) = ( F ` u ) )

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)