Theorem fun 5083

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 5025	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )
2	1	expcom 114	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( F Fn A /\ G Fn B ) -> ( F u. G ) Fn ( A u. B ) ) )
3		rnun 4752	. . . . . 6 |- ran ( F u. G ) = ( ran F u. ran G )
4		unss12 3144	. . . . . 6 |- ( ( ran F C_ C /\ ran G C_ D ) -> ( ran F u. ran G ) C_ ( C u. D ) )
5	3, 4	syl5eqss 3043	. . . . 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 328	. . 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 4926	. . . . 5 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
9		df-f 4926	. . . . 5 |- ( G : B --> D <-> ( G Fn B /\ ran G C_ D ) )
10	8, 9	anbi12i 447	. . . 4 |- ( ( F : A --> C /\ G : B --> D ) <-> ( ( F Fn A /\ ran F C_ C ) /\ ( G Fn B /\ ran G C_ D ) ) )
11		an4 550	. . . 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 182	. . 3 |- ( ( F : A --> C /\ G : B --> D ) <-> ( ( F Fn A /\ G Fn B ) /\ ( ran F C_ C /\ ran G C_ D ) ) )
13		df-f 4926	. . 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 203	. 2 |- ( ( A i^i B ) = (/) -> ( ( F : A --> C /\ G : B --> D ) -> ( F u. G ) : ( A u. B ) --> ( C u. D ) ) )
15	14	impcom 123	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 102   = wceq 1284   u. cun 2971   i^i cin 2972   C_ wss 2973   (/)c0 3251   ran crn 4364   Fn wfn 4917   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926

This theorem is referenced by:  fun2  5084  ftpg  5368  fsnunf  5383