Theorem fun 5470

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 5402	. . . . 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 5111	. . . . . 6 |- ran ( F u. G ) = ( ran F u. ran G )
4		unss12 3354	. . . . . 6 |- ( ( ran F C_ C /\ ran G C_ D ) -> ( ran F u. ran G ) C_ ( C u. D ) )
5	3, 4	eqsstrid 3248	. . . . 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 5295	. . . . 5 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
9		df-f 5295	. . . . 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 5295	. . 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 1373   u. cun 3173   i^i cin 3174   C_ wss 3175   (/)c0 3469   ran crn 4695   Fn wfn 5286   -->wf 5287

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2779  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-br 4061  df-opab 4123  df-id 4359  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-fun 5293  df-fn 5294  df-f 5295

This theorem is referenced by:  fun2  5471  ftpg  5793  fsnunf  5809  cats1un  11214