Theorem fmbas 10311

Description: The base set of a mapping filter is the first argument. (Contributed by Jeff Hankins, 18-Sep-2009.)

Hypothesis

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

Assertion

Ref	Expression
fmbas	|- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U.((X FilMap B)` F) = X)

Proof of Theorem fmbas

Step	Hyp	Ref	Expression
1		fmbas.1	. . . 4 |- Y = U.B
2	1	isfilmap 10308	. . 3 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> ((X FilMap B)` F) = (filGen` ({w | E.t e. B w = (F"t)} u. {X})))
3	2	unieqd 3188	. 2 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U.((X FilMap B)` F) = U.(filGen` ({w | E.t e. B w = (F"t)} u. {X})))
4		eqid 1884	. . . . . . 7 |- {w | E.t e. B w = (F"t)} = {w | E.t e. B w = (F"t)}
5	1, 4	filrn 10293	. . . . . 6 |- ((B e. fBas /\ F Fn Y) -> {w | E.t e. B w = (F"t)} e. fBas)
6		ffn 4562	. . . . . 6 |- (F:Y-->X -> F Fn Y)
7	5, 6	sylan2 500	. . . . 5 |- ((B e. fBas /\ F:Y-->X) -> {w | E.t e. B w = (F"t)} e. fBas)
8	7	3adant1 894	. . . 4 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> {w | E.t e. B w = (F"t)} e. fBas)
9		ffun 4565	. . . . . . . . . 10 |- (F:Y-->X -> Fun F)
10		visset 2295	. . . . . . . . . . 11 |- t e. _V
11	10	funimaex 4496	. . . . . . . . . 10 |- (Fun F -> (F"t) e. _V)
12	9, 11	syl 12	. . . . . . . . 9 |- (F:Y-->X -> (F"t) e. _V)
13	12	a1d 15	. . . . . . . 8 |- (F:Y-->X -> (t e. B -> (F"t) e. _V))
14	13	r19.21aiv 2175	. . . . . . 7 |- (F:Y-->X -> A.t e. B (F"t) e. _V)
15	14	3ad2ant3 899	. . . . . 6 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> A.t e. B (F"t) e. _V)
16		dfiun2g 3283	. . . . . 6 |- (A.t e. B (F"t) e. _V -> U_t e. B (F"t) = U.{w | E.t e. B w = (F"t)})
17	15, 16	syl 12	. . . . 5 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U_t e. B (F"t) = U.{w | E.t e. B w = (F"t)})
18		imassrn 4278	. . . . . . . . . . 11 |- (F"t) C_ ran F
19	18	a1i 8	. . . . . . . . . 10 |- (F:Y-->X -> (F"t) C_ ran F)
20		frn 4569	. . . . . . . . . 10 |- (F:Y-->X -> ran F C_ X)
21	19, 20	sstrd 2627	. . . . . . . . 9 |- (F:Y-->X -> (F"t) C_ X)
22	21	a1d 15	. . . . . . . 8 |- (F:Y-->X -> (t e. B -> (F"t) C_ X))
23	22	r19.21aiv 2175	. . . . . . 7 |- (F:Y-->X -> A.t e. B (F"t) C_ X)
24	23	3ad2ant3 899	. . . . . 6 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> A.t e. B (F"t) C_ X)
25		iunss 3291	. . . . . 6 |- (U_t e. B (F"t) C_ X <-> A.t e. B (F"t) C_ X)
26	24, 25	sylibr 217	. . . . 5 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U_t e. B (F"t) C_ X)
27	17, 26	eqsstr3d 2652	. . . 4 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U.{w | E.t e. B w = (F"t)} C_ X)
28		eqid 1884	. . . . 5 |- U.{w | E.t e. B w = (F"t)} = U.{w | E.t e. B w = (F"t)}
29	28	extbas1 10291	. . . 4 |- (({w | E.t e. B w = (F"t)} e. fBas /\ U.{w | E.t e. B w = (F"t)} C_ X) -> ({w | E.t e. B w = (F"t)} u. {X}) e. fBas)
30	8, 27, 29	syl11anc 524	. . 3 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> ({w | E.t e. B w = (F"t)} u. {X}) e. fBas)
31		eqid 1884	. . . 4 |- U.({w | E.t e. B w = (F"t)} u. {X}) = U.({w | E.t e. B w = (F"t)} u. {X})
32	31	fgbas 10286	. . 3 |- (({w | E.t e. B w = (F"t)} u. {X}) e. fBas -> U.({w | E.t e. B w = (F"t)} u. {X}) = U.(filGen` ({w | E.t e. B w = (F"t)} u. {X})))
33	30, 32	syl 12	. 2 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U.({w | E.t e. B w = (F"t)} u. {X}) = U.(filGen` ({w | E.t e. B w = (F"t)} u. {X})))
34		simp1 876	. . 3 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> X e. A)
35	28	extbas2 10292	. . 3 |- ((U.{w | E.t e. B w = (F"t)} C_ X /\ X e. A) -> U.({w | E.t e. B w = (F"t)} u. {X}) = X)
36	27, 34, 35	syl11anc 524	. 2 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U.({w | E.t e. B w = (F"t)} u. {X}) = X)
37	3, 33, 36	3eqtr2d 1932	1 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> U.((X FilMap B)` F) = X)

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  {csn 3044  U.cuni 3177  U_ciun 3255  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  fBascfbas 10257  filGencfg 10258   FilMap cfilmap 10304

This theorem is referenced by:  flimfnei 10319  isflimf 10323  holimf 10326  cnpfillim4 14947  fmufil 15599  fcluscnplem 15617  fcluscnp 15618  isfclusf 15625  flfssfcf 15629  uffcfflf 15630

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-nel 2020  df-ral 2109  df-rex 2110  df-rab 2112  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-iun 3257  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-fbas 10259  df-fg 10260  df-filmap 10306

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