Theorem fun 5430

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 5364	. . . . 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 5078	. . . . . 6 |- ran ( F u. G ) = ( ran F u. ran G )
4		unss12 3335	. . . . . 6 |- ( ( ran F C_ C /\ ran G C_ D ) -> ( ran F u. ran G ) C_ ( C u. D ) )
5	3, 4	eqsstrid 3229	. . . . 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 5262	. . . . 5 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
9		df-f 5262	. . . . 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 5262	. . 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 1364   u. cun 3155   i^i cin 3156   C_ wss 3157   (/)c0 3450   ran crn 4664   Fn wfn 5253   -->wf 5254

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262

This theorem is referenced by:  fun2  5431  ftpg  5746  fsnunf  5762