Theorem fvco4 5493

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 5492	. . 3 |- ( ( K : A --> X /\ u e. A ) -> ( ( H o. K ) ` u ) = ( H ` ( K ` u ) ) )
2	1	ad2ant2r 500	. 2 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( ( H o. K ) ` u ) = ( H ` ( K ` u ) ) )
3		simplr 519	. . . 4 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( H o. K ) = F )
4	3	eqcomd 2145	. . 3 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> F = ( H o. K ) )
5	4	fveq1d 5423	. 2 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( F ` u ) = ( ( H o. K ) ` u ) )
6		fveq2 5421	. . 3 |- ( x = ( K ` u ) -> ( H ` x ) = ( H ` ( K ` u ) ) )
7	6	ad2antll 482	. 2 |- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( H ` x ) = ( H ` ( K ` u ) ) )
8	2, 5, 7	3eqtr4rd 2183	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 1331   e. wcel 1480   o. ccom 4543   -->wf 5119   ` cfv 5123

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131

This theorem is referenced by: (None)