Theorem fun2d 5543

Description: The union of functions with disjoint domains is a function, deduction version of fun2 5542. (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 5542	. 2 |- ( ( ( F : A --> C /\ G : B --> C ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> C )
5	1, 2, 3, 4	syl21anc 1273	1 |- ( ph -> ( F u. G ) : ( A u. B ) --> C )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3212   i^i cin 3213   (/)c0 3512   -->wf 5353

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361

This theorem is referenced by:  mapunen  7117  uhgrun  16193  upgrun  16233  umgrun  16235