Theorem fvun1 5495

Description: The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Assertion

Ref	Expression
fvun1	|- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> ( ( F u. G ) ` X ) = ( F ` X ) )

Proof of Theorem fvun1

Step	Hyp	Ref	Expression
1		fnfun 5228	. . 3 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 1003	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> Fun F )
3		fnfun 5228	. . 3 |- ( G Fn B -> Fun G )
4	3	3ad2ant2 1004	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> Fun G )
5		fndm 5230	. . . . . . 7 |- ( F Fn A -> dom F = A )
6		fndm 5230	. . . . . . 7 |- ( G Fn B -> dom G = B )
7	5, 6	ineqan12d 3284	. . . . . 6 |- ( ( F Fn A /\ G Fn B ) -> ( dom F i^i dom G ) = ( A i^i B ) )
8	7	eqeq1d 2149	. . . . 5 |- ( ( F Fn A /\ G Fn B ) -> ( ( dom F i^i dom G ) = (/) <-> ( A i^i B ) = (/) ) )
9	8	biimprd 157	. . . 4 |- ( ( F Fn A /\ G Fn B ) -> ( ( A i^i B ) = (/) -> ( dom F i^i dom G ) = (/) ) )
10	9	adantrd 277	. . 3 |- ( ( F Fn A /\ G Fn B ) -> ( ( ( A i^i B ) = (/) /\ X e. A ) -> ( dom F i^i dom G ) = (/) ) )
11	10	3impia 1179	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> ( dom F i^i dom G ) = (/) )
12		simp3r 1011	. . 3 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> X e. A )
13	5	eleq2d 2210	. . . 4 |- ( F Fn A -> ( X e. dom F <-> X e. A ) )
14	13	3ad2ant1 1003	. . 3 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> ( X e. dom F <-> X e. A ) )
15	12, 14	mpbird 166	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> X e. dom F )
16		funun 5175	. . . . . . 7 |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
17		ssun1 3244	. . . . . . . . 9 |- F C_ ( F u. G )
18		dmss 4746	. . . . . . . . 9 |- ( F C_ ( F u. G ) -> dom F C_ dom ( F u. G ) )
19	17, 18	ax-mp 5	. . . . . . . 8 |- dom F C_ dom ( F u. G )
20	19	sseli 3098	. . . . . . 7 |- ( X e. dom F -> X e. dom ( F u. G ) )
21	16, 20	anim12i 336	. . . . . 6 |- ( ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) /\ X e. dom F ) -> ( Fun ( F u. G ) /\ X e. dom ( F u. G ) ) )
22	21	anasss 397	. . . . 5 |- ( ( ( Fun F /\ Fun G ) /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( Fun ( F u. G ) /\ X e. dom ( F u. G ) ) )
23	22	3impa 1177	. . . 4 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( Fun ( F u. G ) /\ X e. dom ( F u. G ) ) )
24		funfvdm 5492	. . . 4 |- ( ( Fun ( F u. G ) /\ X e. dom ( F u. G ) ) -> ( ( F u. G ) ` X ) = U. ( ( F u. G ) " { X } ) )
25	23, 24	syl 14	. . 3 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( ( F u. G ) ` X ) = U. ( ( F u. G ) " { X } ) )
26		imaundir 4960	. . . . . 6 |- ( ( F u. G ) " { X } ) = ( ( F " { X } ) u. ( G " { X } ) )
27	26	a1i 9	. . . . 5 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( ( F u. G ) " { X } ) = ( ( F " { X } ) u. ( G " { X } ) ) )
28	27	unieqd 3755	. . . 4 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> U. ( ( F u. G ) " { X } ) = U. ( ( F " { X } ) u. ( G " { X } ) ) )
29		disjel 3422	. . . . . . . . 9 |- ( ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) -> -. X e. dom G )
30		ndmima 4924	. . . . . . . . 9 |- ( -. X e. dom G -> ( G " { X } ) = (/) )
31	29, 30	syl 14	. . . . . . . 8 |- ( ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) -> ( G " { X } ) = (/) )
32	31	3ad2ant3 1005	. . . . . . 7 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( G " { X } ) = (/) )
33	32	uneq2d 3235	. . . . . 6 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( ( F " { X } ) u. ( G " { X } ) ) = ( ( F " { X } ) u. (/) ) )
34		un0 3401	. . . . . 6 |- ( ( F " { X } ) u. (/) ) = ( F " { X } )
35	33, 34	eqtrdi 2189	. . . . 5 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( ( F " { X } ) u. ( G " { X } ) ) = ( F " { X } ) )
36	35	unieqd 3755	. . . 4 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> U. ( ( F " { X } ) u. ( G " { X } ) ) = U. ( F " { X } ) )
37	28, 36	eqtrd 2173	. . 3 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> U. ( ( F u. G ) " { X } ) = U. ( F " { X } ) )
38		funfvdm 5492	. . . . . 6 |- ( ( Fun F /\ X e. dom F ) -> ( F ` X ) = U. ( F " { X } ) )
39	38	eqcomd 2146	. . . . 5 |- ( ( Fun F /\ X e. dom F ) -> U. ( F " { X } ) = ( F ` X ) )
40	39	adantrl 470	. . . 4 |- ( ( Fun F /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> U. ( F " { X } ) = ( F ` X ) )
41	40	3adant2 1001	. . 3 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> U. ( F " { X } ) = ( F ` X ) )
42	25, 37, 41	3eqtrd 2177	. 2 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( ( F u. G ) ` X ) = ( F ` X ) )
43	2, 4, 11, 15, 42	syl112anc 1221	1 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> ( ( F u. G ) ` X ) = ( F ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   u. cun 3074   i^i cin 3075   C_ wss 3076   (/)c0 3368   {csn 3532   U.cuni 3744   dom cdm 4547   "cima 4550   Fun wfun 5125   Fn wfn 5126   ` cfv 5131

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by:  fvun2  5496  caseinl  6984