Theorem fnun 5484

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 5375	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		df-fn 5375	. . 3 |- ( G Fn B <-> ( Fun G /\ dom G = B ) )
3		ineq12 3427	. . . . . . . . . . 11 |- ( ( dom F = A /\ dom G = B ) -> ( dom F i^i dom G ) = ( A i^i B ) )
4	3	eqeq1d 2247	. . . . . . . . . 10 |- ( ( dom F = A /\ dom G = B ) -> ( ( dom F i^i dom G ) = (/) <-> ( A i^i B ) = (/) ) )
5	4	anbi2d 468	. . . . . . . . 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 5417	. . . . . . . . 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 4983	. . . . . . . . 9 |- dom ( F u. G ) = ( dom F u. dom G )
9		uneq12 3378	. . . . . . . . 9 |- ( ( dom F = A /\ dom G = B ) -> ( dom F u. dom G ) = ( A u. B ) )
10	8, 9	eqtrid 2283	. . . . . . . 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 5375	. . . . . . 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 596	. . 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 1402   u. cun 3218   i^i cin 3219   (/)c0 3520   dom cdm 4769   Fun wfun 5366   Fn wfn 5367

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-fun 5374  df-fn 5375

This theorem is referenced by:  fnunsn  5485  fun  5556  foun  5653  f1oun  5654  xnn0nnen  10852