Theorem fnun 5445

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 5336	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		df-fn 5336	. . 3 |- ( G Fn B <-> ( Fun G /\ dom G = B ) )
3		ineq12 3405	. . . . . . . . . . 11 |- ( ( dom F = A /\ dom G = B ) -> ( dom F i^i dom G ) = ( A i^i B ) )
4	3	eqeq1d 2240	. . . . . . . . . 10 |- ( ( dom F = A /\ dom G = B ) -> ( ( dom F i^i dom G ) = (/) <-> ( A i^i B ) = (/) ) )
5	4	anbi2d 464	. . . . . . . . 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 5378	. . . . . . . . 9 |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
7	5, 6	biimtrrdi 164	. . . . . . . 8 |- ( ( dom F = A /\ dom G = B ) -> ( ( ( Fun F /\ Fun G ) /\ ( A i^i B ) = (/) ) -> Fun ( F u. G ) ) )
8		dmun 4944	. . . . . . . . 9 |- dom ( F u. G ) = ( dom F u. dom G )
9		uneq12 3358	. . . . . . . . 9 |- ( ( dom F = A /\ dom G = B ) -> ( dom F u. dom G ) = ( A u. B ) )
10	8, 9	eqtrid 2276	. . . . . . . 8 |- ( ( dom F = A /\ dom G = B ) -> dom ( F u. G ) = ( A u. B ) )
11	7, 10	jctird 317	. . . . . . 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 5336	. . . . . . 7 |- ( ( F u. G ) Fn ( A u. B ) <-> ( Fun ( F u. G ) /\ dom ( F u. G ) = ( A u. B ) ) )
13	11, 12	imbitrrdi 162	. . . . . 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 258	. . . . 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 125	. . . 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 592	. . 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 291	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( ( A i^i B ) = (/) -> ( F u. G ) Fn ( A u. B ) ) )
18	17	imp 124	1 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   u. cun 3199   i^i cin 3200   (/)c0 3496   dom cdm 4731   Fun wfun 5327   Fn wfn 5328

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-fun 5335  df-fn 5336

This theorem is referenced by:  fnunsn  5446  fun  5516  foun  5611  f1oun  5612  xnn0nnen  10745