Metamath Proof Explorer


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 ) )