Theorem fvco4 5552

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 5551	. . 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 2170	. . 3 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> F = ( H o. K ) )
5	4	fveq1d 5482	. 2 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( F ` u ) = ( ( H o. K ) ` u ) )
6		fveq2 5480	. . 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 2208	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 1342   e. wcel 2135   o. ccom 4602   -->wf 5178   ` cfv 5182

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190

This theorem is referenced by: (None)