Theorem fun2d 5510

Description: The union of functions with disjoint domains is a function, deduction version of fun2 5509. (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 5509	. 2 |- ( ( ( F : A --> C /\ G : B --> C ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> C )
5	1, 2, 3, 4	syl21anc 1272	1 |- ( ph -> ( F u. G ) : ( A u. B ) --> C )

Syntax hints:   -> wi 4   = wceq 1397   u. cun 3198   i^i cin 3199   (/)c0 3494   -->wf 5322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by:  uhgrun  15943  upgrun  15983  umgrun  15985