Theorem fun2d 5537

Description: The union of functions with disjoint domains is a function, deduction version of fun2 5536. (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 5536	. 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 3208   i^i cin 3209   (/)c0 3507   -->wf 5347

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 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-id 4413  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-fun 5353  df-fn 5354  df-f 5355

This theorem is referenced by:  mapunen  7103  uhgrun  16073  upgrun  16113  umgrun  16115