Theorem fnun 5025

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

Assertion

Ref	Expression
fnun	|- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )

Proof of Theorem fnun

Step	Hyp	Ref	Expression
1		df-fn 4925	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		df-fn 4925	. . 3 |- ( G Fn B <-> ( Fun G /\ dom G = B ) )
3		ineq12 3162	. . . . . . . . . . 11 |- ( ( dom F = A /\ dom G = B ) -> ( dom F i^i dom G ) = ( A i^i B ) )
4	3	eqeq1d 2089	. . . . . . . . . 10 |- ( ( dom F = A /\ dom G = B ) -> ( ( dom F i^i dom G ) = (/) <-> ( A i^i B ) = (/) ) )
5	4	anbi2d 451	. . . . . . . . 9 |- ( ( dom F = A /\ dom G = B ) -> ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) <-> ( ( Fun F /\ Fun G ) /\ ( A i^i B ) = (/) ) ) )
6		funun 4964	. . . . . . . . 9 |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
7	5, 6	syl6bir 162	. . . . . . . 8 |- ( ( dom F = A /\ dom G = B ) -> ( ( ( Fun F /\ Fun G ) /\ ( A i^i B ) = (/) ) -> Fun ( F u. G ) ) )
8		dmun 4560	. . . . . . . . 9 |- dom ( F u. G ) = ( dom F u. dom G )
9		uneq12 3121	. . . . . . . . 9 |- ( ( dom F = A /\ dom G = B ) -> ( dom F u. dom G ) = ( A u. B ) )
10	8, 9	syl5eq 2125	. . . . . . . 8 |- ( ( dom F = A /\ dom G = B ) -> dom ( F u. G ) = ( A u. B ) )
11	7, 10	jctird 310	. . . . . . 7 |- ( ( dom F = A /\ dom G = B ) -> ( ( ( Fun F /\ Fun G ) /\ ( A i^i B ) = (/) ) -> ( Fun ( F u. G ) /\ dom ( F u. G ) = ( A u. B ) ) ) )
12		df-fn 4925	. . . . . . 7 |- ( ( F u. G ) Fn ( A u. B ) <-> ( Fun ( F u. G ) /\ dom ( F u. G ) = ( A u. B ) ) )
13	11, 12	syl6ibr 160	. . . . . 6 |- ( ( dom F = A /\ dom G = B ) -> ( ( ( Fun F /\ Fun G ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) ) )
14	13	expd 254	. . . . 5 |- ( ( dom F = A /\ dom G = B ) -> ( ( Fun F /\ Fun G ) -> ( ( A i^i B ) = (/) -> ( F u. G ) Fn ( A u. B ) ) ) )
15	14	impcom 123	. . . 4 |- ( ( ( Fun F /\ Fun G ) /\ ( dom F = A /\ dom G = B ) ) -> ( ( A i^i B ) = (/) -> ( F u. G ) Fn ( A u. B ) ) )
16	15	an4s 552	. . 3 |- ( ( ( Fun F /\ dom F = A ) /\ ( Fun G /\ dom G = B ) ) -> ( ( A i^i B ) = (/) -> ( F u. G ) Fn ( A u. B ) ) )
17	1, 2, 16	syl2anb 285	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( ( A i^i B ) = (/) -> ( F u. G ) Fn ( A u. B ) ) )
18	17	imp 122	1 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   u. cun 2971   i^i cin 2972   (/)c0 3251   dom cdm 4363   Fun wfun 4916   Fn wfn 4917

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-fun 4924  df-fn 4925

This theorem is referenced by:  fnunsn  5026  fun  5083  foun  5165  f1oun  5166