Theorem fun2d 6068

Description: The union of functions with disjoint domains is a function, deduction version of fun2 6067. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
fun2d.f	|- ( ph -> F : A --> C )
fun2d.g	|- ( ph -> G : B --> C )
fun2d.i	|- ( ph -> ( A i^i B ) = (/) )

Assertion

Ref	Expression
fun2d	|- ( ph -> ( F u. G ) : ( A u. B ) --> C )

Proof of Theorem fun2d

Step	Hyp	Ref	Expression
1		fun2d.f	. 2 |- ( ph -> F : A --> C )
2		fun2d.g	. 2 |- ( ph -> G : B --> C )
3		fun2d.i	. 2 |- ( ph -> ( A i^i B ) = (/) )
4		fun2 6067	. 2 |- ( ( ( F : A --> C /\ G : B --> C ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> C )
5	1, 2, 3, 4	syl21anc 1325	1 |- ( ph -> ( F u. G ) : ( A u. B ) --> C )

Syntax hints:   -> wi 4   = wceq 1483   u. cun 3572   i^i cin 3573   (/)c0 3915   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  uhgrun  25969  upgrun  26013  umgrun  26015  reprsuc  30693