Theorem fun2d 5561

Description: The union of functions with disjoint domains is a function, deduction version of fun2 5560. (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 5560	. 2 |- ( ( ( F : A --> C /\ G : B --> C ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> C )
5	1, 2, 3, 4	syl21anc 1277	1 |- ( ph -> ( F u. G ) : ( A u. B ) --> C )

Syntax hints:   -> wi 4   = wceq 1402   u. cun 3218   i^i cin 3219   (/)c0 3520   -->wf 5371

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-id 4436  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-fun 5377  df-fn 5378  df-f 5379

This theorem is referenced by:  mapunen  7145  uhgrun  16310  upgrun  16350  umgrun  16352