Theorem filmapf 10307

Description: Given a function from a filtered set to a topology, return the filter of supersets of images of filter elements. (Contributed by Jeff Hankins, 5-Sep-2009.)

Hypothesis

Ref	Expression
filmapf.1	|- Y = U.B

Assertion

Ref	Expression
filmapf	|- ((X e. A /\ B e. fBas) -> (X FilMap B) = {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))})

Distinct variable groups:   f,s,A   t,f,w,B,s   f,X,s   f,Y,s

Proof of Theorem filmapf

Step	Hyp	Ref	Expression
1		elmapg 5392	. . . . . 6 |- ((X e. A /\ Y e. _V) -> (f e. (X ^m Y) <-> f:Y-->X))
2		uniexg 3795	. . . . . . 7 |- (B e. fBas -> U.B e. _V)
3		filmapf.1	. . . . . . 7 |- Y = U.B
4	2, 3	syl5eqel 1975	. . . . . 6 |- (B e. fBas -> Y e. _V)
5	1, 4	sylan2 500	. . . . 5 |- ((X e. A /\ B e. fBas) -> (f e. (X ^m Y) <-> f:Y-->X))
6	5	anbi1d 679	. . . 4 |- ((X e. A /\ B e. fBas) -> ((f e. (X ^m Y) /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X}))) <-> (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))))
7	6	opabbidv 3401	. . 3 |- ((X e. A /\ B e. fBas) -> {<.f, s>. | (f e. (X ^m Y) /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))} = {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))})
8		oprex 4907	. . . 4 |- (X ^m Y) e. _V
9	8	opabex2 4539	. . 3 |- {<.f, s>. | (f e. (X ^m Y) /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))} e. _V
10	7, 9	syl6eqelr 1980	. 2 |- ((X e. A /\ B e. fBas) -> {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))} e. _V)
11		feq3 4553	. . . . . 6 |- (x = X -> (f:U.y-->x <-> f:U.y-->X))
12		sneq 3054	. . . . . . . . 9 |- (x = X -> {x} = {X})
13	12	uneq2d 2755	. . . . . . . 8 |- (x = X -> ({w | E.t e. y w = (f"t)} u. {x}) = ({w | E.t e. y w = (f"t)} u. {X}))
14	13	fveq2d 4685	. . . . . . 7 |- (x = X -> (filGen` ({w | E.t e. y w = (f"t)} u. {x})) = (filGen` ({w | E.t e. y w = (f"t)} u. {X})))
15	14	eqeq2d 1895	. . . . . 6 |- (x = X -> (s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})) <-> s = (filGen` ({w | E.t e. y w = (f"t)} u. {X}))))
16	11, 15	anbi12d 690	. . . . 5 |- (x = X -> ((f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x}))) <-> (f:U.y-->X /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {X})))))
17	16	opabbidv 3401	. . . 4 |- (x = X -> {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))} = {<.f, s>. | (f:U.y-->X /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {X})))})
18		unieq 3185	. . . . . . . 8 |- (y = B -> U.y = U.B)
19	18, 3	syl6eqr 1946	. . . . . . 7 |- (y = B -> U.y = Y)
20	19	feq2d 4557	. . . . . 6 |- (y = B -> (f:U.y-->X <-> f:Y-->X))
21		rexeq 2267	. . . . . . . . . 10 |- (y = B -> (E.t e. y w = (f"t) <-> E.t e. B w = (f"t)))
22	21	abbidv 2008	. . . . . . . . 9 |- (y = B -> {w | E.t e. y w = (f"t)} = {w | E.t e. B w = (f"t)})
23	22	uneq1d 2754	. . . . . . . 8 |- (y = B -> ({w | E.t e. y w = (f"t)} u. {X}) = ({w | E.t e. B w = (f"t)} u. {X}))
24	23	fveq2d 4685	. . . . . . 7 |- (y = B -> (filGen` ({w | E.t e. y w = (f"t)} u. {X})) = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))
25	24	eqeq2d 1895	. . . . . 6 |- (y = B -> (s = (filGen` ({w | E.t e. y w = (f"t)} u. {X})) <-> s = (filGen` ({w | E.t e. B w = (f"t)} u. {X}))))
26	20, 25	anbi12d 690	. . . . 5 |- (y = B -> ((f:U.y-->X /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {X}))) <-> (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))))
27	26	opabbidv 3401	. . . 4 |- (y = B -> {<.f, s>. | (f:U.y-->X /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {X})))} = {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))})
28		df-filmap 10306	. . . . 5 |- FilMap = {<.<.x, y>., z>. | (y e. fBas /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))})}
29		visset 2295	. . . . . . . 8 |- x e. _V
30	29	biantrur 794	. . . . . . 7 |- ((y e. fBas /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))}) <-> (x e. _V /\ (y e. fBas /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))})))
31		anass 487	. . . . . . 7 |- (((x e. _V /\ y e. fBas) /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))}) <-> (x e. _V /\ (y e. fBas /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))})))
32	30, 31	bitr4i 193	. . . . . 6 |- ((y e. fBas /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))}) <-> ((x e. _V /\ y e. fBas) /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))}))
33	32	oprabbii 4923	. . . . 5 |- {<.<.x, y>., z>. | (y e. fBas /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))})} = {<.<.x, y>., z>. | ((x e. _V /\ y e. fBas) /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))})}
34	28, 33	eqtri 1908	. . . 4 |- FilMap = {<.<.x, y>., z>. | ((x e. _V /\ y e. fBas) /\ z = {<.f, s>. | (f:U.y-->x /\ s = (filGen` ({w | E.t e. y w = (f"t)} u. {x})))})}
35	17, 27, 34	oprabval2g 4956	. . 3 |- ((X e. _V /\ B e. fBas /\ {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))} e. _V) -> (X FilMap B) = {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))})
36		elisset 2299	. . 3 |- (X e. A -> X e. _V)
37	35, 36	syl3an1 1130	. 2 |- ((X e. A /\ B e. fBas /\ {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))} e. _V) -> (X FilMap B) = {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))})
38	10, 37	mpd3an3 1192	1 |- ((X e. A /\ B e. fBas) -> (X FilMap B) = {<.f, s>. | (f:Y-->X /\ s = (filGen` ({w | E.t e. B w = (f"t)} u. {X})))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292   u. cun 2591  {csn 3044  U.cuni 3177  {copab 3395  "cima 3989  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885   ^m cmap 5381  fBascfbas 10257  filGencfg 10258   FilMap cfilmap 10304

This theorem is referenced by:  isfilmap 10308

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-filmap 10306

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