Theorem fnund

Description: The union of two functions with disjoint domains, a deduction version. (Contributed by metakunt, 28-May-2024)

	Ref	Expression
Hypotheses	fnund.1	|- ( ph -> F Fn A )
	fnund.2	|- ( ph -> G Fn B )
	fnund.3	|- ( ph -> ( A i^i B ) = (/) )
Assertion	fnund	|- ( ph -> ( F u. G ) Fn ( A u. B ) )

Proof

Step	Hyp	Ref	Expression
1		fnund.1	|- ( ph -> F Fn A )
2		fnund.2	|- ( ph -> G Fn B )
3		fnund.3	|- ( ph -> ( A i^i B ) = (/) )
4		fnun	|- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )
5	1 2 3 4	syl21anc	|- ( ph -> ( F u. G ) Fn ( A u. B ) )