Theorem fnun 5229

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 5126	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		df-fn 5126	. . 3 |- ( G Fn B <-> ( Fun G /\ dom G = B ) )
3		ineq12 3272	. . . . . . . . . . 11 |- ( ( dom F = A /\ dom G = B ) -> ( dom F i^i dom G ) = ( A i^i B ) )
4	3	eqeq1d 2148	. . . . . . . . . 10 |- ( ( dom F = A /\ dom G = B ) -> ( ( dom F i^i dom G ) = (/) <-> ( A i^i B ) = (/) ) )
5	4	anbi2d 459	. . . . . . . . 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 5167	. . . . . . . . 9 |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
7	5, 6	syl6bir 163	. . . . . . . 8 |- ( ( dom F = A /\ dom G = B ) -> ( ( ( Fun F /\ Fun G ) /\ ( A i^i B ) = (/) ) -> Fun ( F u. G ) ) )
8		dmun 4746	. . . . . . . . 9 |- dom ( F u. G ) = ( dom F u. dom G )
9		uneq12 3225	. . . . . . . . 9 |- ( ( dom F = A /\ dom G = B ) -> ( dom F u. dom G ) = ( A u. B ) )
10	8, 9	syl5eq 2184	. . . . . . . 8 |- ( ( dom F = A /\ dom G = B ) -> dom ( F u. G ) = ( A u. B ) )
11	7, 10	jctird 315	. . . . . . 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 5126	. . . . . . 7 |- ( ( F u. G ) Fn ( A u. B ) <-> ( Fun ( F u. G ) /\ dom ( F u. G ) = ( A u. B ) ) )
13	11, 12	syl6ibr 161	. . . . . 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 256	. . . . 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 124	. . . 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 577	. . 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 289	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( ( A i^i B ) = (/) -> ( F u. G ) Fn ( A u. B ) ) )
18	17	imp 123	1 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   u. cun 3069   i^i cin 3070   (/)c0 3363   dom cdm 4539   Fun wfun 5117   Fn wfn 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-fun 5125  df-fn 5126

This theorem is referenced by:  fnunsn  5230  fun  5295  foun  5386  f1oun  5387