Theorem fvco4 5558

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 5557	. . 3 |- ( ( K : A --> X /\ u e. A ) -> ( ( H o. K ) ` u ) = ( H ` ( K ` u ) ) )
2	1	ad2ant2r 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 )
4	3	eqcomd 2171	. . 3 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> F = ( H o. K ) )
5	4	fveq1d 5488	. 2 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( F ` u ) = ( ( H o. K ) ` u ) )
6		fveq2 5486	. . 3 |- ( x = ( K ` u ) -> ( H ` x ) = ( H ` ( K ` u ) ) )
7	6	ad2antll 483	. 2 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( H ` x ) = ( H ` ( K ` u ) ) )
8	2, 5, 7	3eqtr4rd 2209	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 1343   e. wcel 2136   o. ccom 4608   -->wf 5184   ` cfv 5188

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196

This theorem is referenced by: (None)