Theorem fnunres2

Description: Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023)

	Ref	Expression
Assertion	fnunres2	|- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G )

Proof

Step	Hyp	Ref	Expression
1		uncom	|- ( F u. G ) = ( G u. F )
2	1	reseq1i	|- ( ( F u. G ) |` B ) = ( ( G u. F ) |` B )
3		ineqcom	|- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
4		fnunres1	|- ( ( G Fn B /\ F Fn A /\ ( B i^i A ) = (/) ) -> ( ( G u. F ) |` B ) = G )
5	3 4	syl3an3b	|- ( ( G Fn B /\ F Fn A /\ ( A i^i B ) = (/) ) -> ( ( G u. F ) |` B ) = G )
6	5	3com12	|- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( G u. F ) |` B ) = G )
7	2 6	eqtrid	|- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G )