Theorem fvun1 5239

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 4996	. . 3 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 925	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> Fun F )
3		fnfun 4996	. . 3 |- ( G Fn B -> Fun G )
4	3	3ad2ant2 926	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> Fun G )
5		fndm 4998	. . . . . . 7 |- ( F Fn A -> dom F = A )
6		fndm 4998	. . . . . . 7 |- ( G Fn B -> dom G = B )
7	5, 6	ineqan12d 3140	. . . . . 6 |- ( ( F Fn A /\ G Fn B ) -> ( dom F i^i dom G ) = ( A i^i B ) )
8	7	eqeq1d 2048	. . . . 5 |- ( ( F Fn A /\ G Fn B ) -> ( ( dom F i^i dom G ) = (/) <-> ( A i^i B ) = (/) ) )
9	8	biimprd 147	. . . 4 |- ( ( F Fn A /\ G Fn B ) -> ( ( A i^i B ) = (/) -> ( dom F i^i dom G ) = (/) ) )
10	9	adantrd 264	. . 3 |- ( ( F Fn A /\ G Fn B ) -> ( ( ( A i^i B ) = (/) /\ X e. A ) -> ( dom F i^i dom G ) = (/) ) )
11	10	3impia 1101	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> ( dom F i^i dom G ) = (/) )
12		simp3r 933	. . 3 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> X e. A )
13	5	eleq2d 2107	. . . 4 |- ( F Fn A -> ( X e. dom F <-> X e. A ) )
14	13	3ad2ant1 925	. . 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 156	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> X e. dom F )
16		funun 4944	. . . . . . 7 |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
17		ssun1 3106	. . . . . . . . 9 |- F C_ ( F u. G )
18		dmss 4534	. . . . . . . . 9 |- ( F C_ ( F u. G ) -> dom F C_ dom ( F u. G ) )
19	17, 18	ax-mp 7	. . . . . . . 8 |- dom F C_ dom ( F u. G )
20	19	sseli 2941	. . . . . . 7 |- ( X e. dom F -> X e. dom ( F u. G ) )
21	16, 20	anim12i 321	. . . . . 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 379	. . . . 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 1099	. . . 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 5236	. . . 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 4737	. . . . . 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 3591	. . . 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 3274	. . . . . . . . 9 |- ( ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) -> -. X e. dom G )
30		ndmima 4702	. . . . . . . . 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 927	. . . . . . 7 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( G " { X } ) = (/) )
33	32	uneq2d 3097	. . . . . 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 3251	. . . . . 6 |- ( ( F " { X } ) u. (/) ) = ( F " { X } )
35	33, 34	syl6eq 2088	. . . . 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 3591	. . . 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 2072	. . 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 5236	. . . . . 6 |- ( ( Fun F /\ X e. dom F ) -> ( F ` X ) = U. ( F " { X } ) )
39	38	eqcomd 2045	. . . . 5 |- ( ( Fun F /\ X e. dom F ) -> U. ( F " { X } ) = ( F ` X ) )
40	39	adantrl 447	. . . 4 |- ( ( Fun F /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> U. ( F " { X } ) = ( F ` X ) )
41	40	3adant2 923	. . 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 2076	. 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 1139	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 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   u. cun 2915   i^i cin 2916   C_ wss 2917   (/)c0 3224   {csn 3375   U.cuni 3580   dom cdm 4345   "cima 4348   Fun wfun 4896   Fn wfn 4897   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by:  fvun2  5240