Theorem fvun1 5627

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 5355	. . 3 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 1020	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> Fun F )
3		fnfun 5355	. . 3 |- ( G Fn B -> Fun G )
4	3	3ad2ant2 1021	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> Fun G )
5		fndm 5357	. . . . . . 7 |- ( F Fn A -> dom F = A )
6		fndm 5357	. . . . . . 7 |- ( G Fn B -> dom G = B )
7	5, 6	ineqan12d 3366	. . . . . 6 |- ( ( F Fn A /\ G Fn B ) -> ( dom F i^i dom G ) = ( A i^i B ) )
8	7	eqeq1d 2205	. . . . 5 |- ( ( F Fn A /\ G Fn B ) -> ( ( dom F i^i dom G ) = (/) <-> ( A i^i B ) = (/) ) )
9	8	biimprd 158	. . . 4 |- ( ( F Fn A /\ G Fn B ) -> ( ( A i^i B ) = (/) -> ( dom F i^i dom G ) = (/) ) )
10	9	adantrd 279	. . 3 |- ( ( F Fn A /\ G Fn B ) -> ( ( ( A i^i B ) = (/) /\ X e. A ) -> ( dom F i^i dom G ) = (/) ) )
11	10	3impia 1202	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> ( dom F i^i dom G ) = (/) )
12		simp3r 1028	. . 3 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> X e. A )
13	5	eleq2d 2266	. . . 4 |- ( F Fn A -> ( X e. dom F <-> X e. A ) )
14	13	3ad2ant1 1020	. . 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 167	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> X e. dom F )
16		funun 5302	. . . . . . 7 |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
17		ssun1 3326	. . . . . . . . 9 |- F C_ ( F u. G )
18		dmss 4865	. . . . . . . . 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 3179	. . . . . . 7 |- ( X e. dom F -> X e. dom ( F u. G ) )
21	16, 20	anim12i 338	. . . . . 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 399	. . . . 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 1196	. . . 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 5624	. . . 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 5083	. . . . . 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 3850	. . . 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 3505	. . . . . . . . 9 |- ( ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) -> -. X e. dom G )
30		ndmima 5046	. . . . . . . . 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 1022	. . . . . . 7 |- ( ( Fun F /\ Fun G /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> ( G " { X } ) = (/) )
33	32	uneq2d 3317	. . . . . 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 3484	. . . . . 6 |- ( ( F " { X } ) u. (/) ) = ( F " { X } )
35	33, 34	eqtrdi 2245	. . . . 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 3850	. . . 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 2229	. . 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 5624	. . . . . 6 |- ( ( Fun F /\ X e. dom F ) -> ( F ` X ) = U. ( F " { X } ) )
39	38	eqcomd 2202	. . . . 5 |- ( ( Fun F /\ X e. dom F ) -> U. ( F " { X } ) = ( F ` X ) )
40	39	adantrl 478	. . . 4 |- ( ( Fun F /\ ( ( dom F i^i dom G ) = (/) /\ X e. dom F ) ) -> U. ( F " { X } ) = ( F ` X ) )
41	40	3adant2 1018	. . 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 2233	. 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 1253	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   u. cun 3155   i^i cin 3156   C_ wss 3157   (/)c0 3450   {csn 3622   U.cuni 3839   dom cdm 4663   "cima 4666   Fun wfun 5252   Fn wfn 5253   ` cfv 5258

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  fvun2  5628  caseinl  7157