Theorem fgf 10283

Description: The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.)

Hypothesis

Ref	Expression
fgf.1	|- X = U.F

Assertion

Ref	Expression
fgf	|- (F e. fBas -> (filGen` F) = {x e. ~PX | (F i^i ~Px) =/= (/)})

Distinct variable groups:   x,F   x,X

Proof of Theorem fgf

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (w = F -> ({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` w) = ({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` F))
2		ineq1 2789	. . . . . . 7 |- (w = F -> (w i^i ~Px) = (F i^i ~Px))
3	2	neeq1d 2028	. . . . . 6 |- (w = F -> ((w i^i ~Px) =/= (/) <-> (F i^i ~Px) =/= (/)))
4	3	rabbidv 2287	. . . . 5 |- (w = F -> {x e. ~PU.w | (w i^i ~Px) =/= (/)} = {x e. ~PU.w | (F i^i ~Px) =/= (/)})
5		unieq 3185	. . . . . . . 8 |- (w = F -> U.w = U.F)
6		pweq 3036	. . . . . . . 8 |- (U.w = U.F -> ~PU.w = ~PU.F)
7	5, 6	syl 12	. . . . . . 7 |- (w = F -> ~PU.w = ~PU.F)
8		fgf.1	. . . . . . . 8 |- X = U.F
9		pweq 3036	. . . . . . . 8 |- (X = U.F -> ~PX = ~PU.F)
10	8, 9	ax-mp 7	. . . . . . 7 |- ~PX = ~PU.F
11	7, 10	syl6eqr 1946	. . . . . 6 |- (w = F -> ~PU.w = ~PX)
12		rabeq 2289	. . . . . 6 |- (~PU.w = ~PX -> {x e. ~PU.w | (F i^i ~Px) =/= (/)} = {x e. ~PX | (F i^i ~Px) =/= (/)})
13	11, 12	syl 12	. . . . 5 |- (w = F -> {x e. ~PU.w | (F i^i ~Px) =/= (/)} = {x e. ~PX | (F i^i ~Px) =/= (/)})
14	4, 13	eqtrd 1925	. . . 4 |- (w = F -> {x e. ~PU.w | (w i^i ~Px) =/= (/)} = {x e. ~PX | (F i^i ~Px) =/= (/)})
15	1, 14	eqeq12d 1899	. . 3 |- (w = F -> (({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` w) = {x e. ~PU.w | (w i^i ~Px) =/= (/)} <-> ({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` F) = {x e. ~PX | (F i^i ~Px) =/= (/)}))
16		uniexg 3795	. . . . 5 |- (w e. fBas -> U.w e. _V)
17		pwexg 3489	. . . . 5 |- (U.w e. _V -> ~PU.w e. _V)
18		rabexg 3460	. . . . 5 |- (~PU.w e. _V -> {x e. ~PU.w | (w i^i ~Px) =/= (/)} e. _V)
19	16, 17, 18	3syl 24	. . . 4 |- (w e. fBas -> {x e. ~PU.w | (w i^i ~Px) =/= (/)} e. _V)
20		ineq1 2789	. . . . . . . 8 |- (y = w -> (y i^i ~Px) = (w i^i ~Px))
21	20	neeq1d 2028	. . . . . . 7 |- (y = w -> ((y i^i ~Px) =/= (/) <-> (w i^i ~Px) =/= (/)))
22	21	rabbidv 2287	. . . . . 6 |- (y = w -> {x e. ~PU.y | (y i^i ~Px) =/= (/)} = {x e. ~PU.y | (w i^i ~Px) =/= (/)})
23		unieq 3185	. . . . . . 7 |- (y = w -> U.y = U.w)
24		pweq 3036	. . . . . . 7 |- (U.y = U.w -> ~PU.y = ~PU.w)
25		rabeq 2289	. . . . . . 7 |- (~PU.y = ~PU.w -> {x e. ~PU.y | (w i^i ~Px) =/= (/)} = {x e. ~PU.w | (w i^i ~Px) =/= (/)})
26	23, 24, 25	3syl 24	. . . . . 6 |- (y = w -> {x e. ~PU.y | (w i^i ~Px) =/= (/)} = {x e. ~PU.w | (w i^i ~Px) =/= (/)})
27	22, 26	eqtrd 1925	. . . . 5 |- (y = w -> {x e. ~PU.y | (y i^i ~Px) =/= (/)} = {x e. ~PU.w | (w i^i ~Px) =/= (/)})
28		eqid 1884	. . . . 5 |- {<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})} = {<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}
29	27, 28	fvopab4g 4742	. . . 4 |- ((w e. fBas /\ {x e. ~PU.w | (w i^i ~Px) =/= (/)} e. _V) -> ({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` w) = {x e. ~PU.w | (w i^i ~Px) =/= (/)})
30	19, 29	mpdan 768	. . 3 |- (w e. fBas -> ({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` w) = {x e. ~PU.w | (w i^i ~Px) =/= (/)})
31	15, 30	vtoclga 2352	. 2 |- (F e. fBas -> ({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` F) = {x e. ~PX | (F i^i ~Px) =/= (/)})
32		df-fg 10260	. . 3 |- filGen = {<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}
33	32	fveq1i 4682	. 2 |- (filGen` F) = ({<.y, z>. | (y e. fBas /\ z = {x e. ~PU.y | (y i^i ~Px) =/= (/)})}` F)
34	31, 33	syl5eq 1940	1 |- (F e. fBas -> (filGen` F) = {x e. ~PX | (F i^i ~Px) =/= (/)})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  {crab 2108  _Vcvv 2292   i^i cin 2592  (/)c0 2875  ~Pcpw 3032  U.cuni 3177  {copab 3395  ` cfv 3998  fBascfbas 10257  filGencfg 10258

This theorem is referenced by:  elfg 10284  neifg 15559

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-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-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-rab 2112  df-v 2294  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-fv 4014  df-fg 10260

Copyright terms: Public domain