Theorem fun 5390

Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Assertion

Ref	Expression
fun	|- ( ( ( F : A --> C /\ G : B --> D ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> ( C u. D ) )

Proof of Theorem fun

Step	Hyp	Ref	Expression
1		fnun 5324	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )
2	1	expcom 116	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( F Fn A /\ G Fn B ) -> ( F u. G ) Fn ( A u. B ) ) )
3		rnun 5039	. . . . . 6 |- ran ( F u. G ) = ( ran F u. ran G )
4		unss12 3309	. . . . . 6 |- ( ( ran F C_ C /\ ran G C_ D ) -> ( ran F u. ran G ) C_ ( C u. D ) )
5	3, 4	eqsstrid 3203	. . . . 5 |- ( ( ran F C_ C /\ ran G C_ D ) -> ran ( F u. G ) C_ ( C u. D ) )
6	5	a1i 9	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( ran F C_ C /\ ran G C_ D ) -> ran ( F u. G ) C_ ( C u. D ) ) )
7	2, 6	anim12d 335	. . 3 |- ( ( A i^i B ) = (/) -> ( ( ( F Fn A /\ G Fn B ) /\ ( ran F C_ C /\ ran G C_ D ) ) -> ( ( F u. G ) Fn ( A u. B ) /\ ran ( F u. G ) C_ ( C u. D ) ) ) )
8		df-f 5222	. . . . 5 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
9		df-f 5222	. . . . 5 |- ( G : B --> D <-> ( G Fn B /\ ran G C_ D ) )
10	8, 9	anbi12i 460	. . . 4 |- ( ( F : A --> C /\ G : B --> D ) <-> ( ( F Fn A /\ ran F C_ C ) /\ ( G Fn B /\ ran G C_ D ) ) )
11		an4 586	. . . 4 |- ( ( ( F Fn A /\ ran F C_ C ) /\ ( G Fn B /\ ran G C_ D ) ) <-> ( ( F Fn A /\ G Fn B ) /\ ( ran F C_ C /\ ran G C_ D ) ) )
12	10, 11	bitri 184	. . 3 |- ( ( F : A --> C /\ G : B --> D ) <-> ( ( F Fn A /\ G Fn B ) /\ ( ran F C_ C /\ ran G C_ D ) ) )
13		df-f 5222	. . 3 |- ( ( F u. G ) : ( A u. B ) --> ( C u. D ) <-> ( ( F u. G ) Fn ( A u. B ) /\ ran ( F u. G ) C_ ( C u. D ) ) )
14	7, 12, 13	3imtr4g 205	. 2 |- ( ( A i^i B ) = (/) -> ( ( F : A --> C /\ G : B --> D ) -> ( F u. G ) : ( A u. B ) --> ( C u. D ) ) )
15	14	impcom 125	1 |- ( ( ( F : A --> C /\ G : B --> D ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> ( C u. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   u. cun 3129   i^i cin 3130   C_ wss 3131   (/)c0 3424   ran crn 4629   Fn wfn 5213   -->wf 5214

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222

This theorem is referenced by:  fun2  5391  ftpg  5702  fsnunf  5718