Theorem elfilmap 10312

Description: An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.)

Hypothesis

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

Assertion

Ref	Expression
elfilmap	|- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (A e. ((X FilMap B)` F) <-> (A C_ X /\ E.x e. B (F"x) C_ A)))

Distinct variable groups:   x,A   x,B   x,C   x,F   x,X   x,Y

Proof of Theorem elfilmap

Step	Hyp	Ref	Expression
1		elfilmap.1	. . . 4 |- Y = U.B
2	1	isfilmap 10308	. . 3 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> ((X FilMap B)` F) = (filGen` ({w | E.t e. B w = (F"t)} u. {X})))
3	2	eleq2d 1964	. 2 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (A e. ((X FilMap B)` F) <-> A e. (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. C /\ 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	3ad2ant3 899	. . . . . . . 8 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (F"t) e. _V)
14	13	a1d 15	. . . . . . 7 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (t e. B -> (F"t) e. _V))
15	14	r19.21aiv 2175	. . . . . 6 |- ((X e. C /\ 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. C /\ 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. C /\ 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. C /\ B e. fBas /\ F:Y-->X) -> U_t e. B (F"t) C_ X)
27	17, 26	eqsstr3d 2652	. . . 4 |- ((X e. C /\ 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. C /\ 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	elfg 10284	. . 3 |- (({w | E.t e. B w = (F"t)} u. {X}) e. fBas -> (A e. (filGen` ({w | E.t e. B w = (F"t)} u. {X})) <-> (A C_ U.({w | E.t e. B w = (F"t)} u. {X}) /\ E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A)))
33	30, 32	syl 12	. 2 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (A e. (filGen` ({w | E.t e. B w = (F"t)} u. {X})) <-> (A C_ U.({w | E.t e. B w = (F"t)} u. {X}) /\ E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A)))
34		simp1 876	. . . . 5 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> X e. C)
35	28	extbas2 10292	. . . . 5 |- ((U.{w | E.t e. B w = (F"t)} C_ X /\ X e. C) -> U.({w | E.t e. B w = (F"t)} u. {X}) = X)
36	27, 34, 35	syl11anc 524	. . . 4 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> U.({w | E.t e. B w = (F"t)} u. {X}) = X)
37	36	sseq2d 2645	. . 3 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (A C_ U.({w | E.t e. B w = (F"t)} u. {X}) <-> A C_ X))
38		simp2 877	. . . . . . . . . . . 12 |- ((F:Y-->X /\ t e. B /\ y = (F"t)) -> t e. B)
39	38	adantr 425	. . . . . . . . . . 11 |- (((F:Y-->X /\ t e. B /\ y = (F"t)) /\ y C_ A) -> t e. B)
40		sseq1 2637	. . . . . . . . . . . . 13 |- (y = (F"t) -> (y C_ A <-> (F"t) C_ A))
41	40	3ad2ant3 899	. . . . . . . . . . . 12 |- ((F:Y-->X /\ t e. B /\ y = (F"t)) -> (y C_ A <-> (F"t) C_ A))
42	41	biimpa 460	. . . . . . . . . . 11 |- (((F:Y-->X /\ t e. B /\ y = (F"t)) /\ y C_ A) -> (F"t) C_ A)
43		imaeq2 4260	. . . . . . . . . . . . 13 |- (x = t -> (F"x) = (F"t))
44	43	sseq1d 2644	. . . . . . . . . . . 12 |- (x = t -> ((F"x) C_ A <-> (F"t) C_ A))
45	44	rcla4ev 2381	. . . . . . . . . . 11 |- ((t e. B /\ (F"t) C_ A) -> E.x e. B (F"x) C_ A)
46	39, 42, 45	syl11anc 524	. . . . . . . . . 10 |- (((F:Y-->X /\ t e. B /\ y = (F"t)) /\ y C_ A) -> E.x e. B (F"x) C_ A)
47	46	3exp1 1084	. . . . . . . . 9 |- (F:Y-->X -> (t e. B -> (y = (F"t) -> (y C_ A -> E.x e. B (F"x) C_ A))))
48	47	3ad2ant3 899	. . . . . . . 8 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (t e. B -> (y = (F"t) -> (y C_ A -> E.x e. B (F"x) C_ A))))
49	48	r19.23adv 2215	. . . . . . 7 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (E.t e. B y = (F"t) -> (y C_ A -> E.x e. B (F"x) C_ A)))
50		simp1 876	. . . . . . . . . . . . . 14 |- ((t e. B /\ F:Y-->X /\ y = X) -> t e. B)
51	50	adantr 425	. . . . . . . . . . . . 13 |- (((t e. B /\ F:Y-->X /\ y = X) /\ y C_ A) -> t e. B)
52		sseq1 2637	. . . . . . . . . . . . . . . 16 |- (y = X -> (y C_ A <-> X C_ A))
53	52	3ad2ant3 899	. . . . . . . . . . . . . . 15 |- ((t e. B /\ F:Y-->X /\ y = X) -> (y C_ A <-> X C_ A))
54		sstr2 2623	. . . . . . . . . . . . . . . . 17 |- ((F"t) C_ X -> (X C_ A -> (F"t) C_ A))
55	21, 54	syl 12	. . . . . . . . . . . . . . . 16 |- (F:Y-->X -> (X C_ A -> (F"t) C_ A))
56	55	3ad2ant2 898	. . . . . . . . . . . . . . 15 |- ((t e. B /\ F:Y-->X /\ y = X) -> (X C_ A -> (F"t) C_ A))
57	53, 56	sylbid 220	. . . . . . . . . . . . . 14 |- ((t e. B /\ F:Y-->X /\ y = X) -> (y C_ A -> (F"t) C_ A))
58	57	imp 377	. . . . . . . . . . . . 13 |- (((t e. B /\ F:Y-->X /\ y = X) /\ y C_ A) -> (F"t) C_ A)
59	51, 58, 45	syl11anc 524	. . . . . . . . . . . 12 |- (((t e. B /\ F:Y-->X /\ y = X) /\ y C_ A) -> E.x e. B (F"x) C_ A)
60	59	3exp1 1084	. . . . . . . . . . 11 |- (t e. B -> (F:Y-->X -> (y = X -> (y C_ A -> E.x e. B (F"x) C_ A))))
61	60	a1i 8	. . . . . . . . . 10 |- (X e. C -> (t e. B -> (F:Y-->X -> (y = X -> (y C_ A -> E.x e. B (F"x) C_ A)))))
62	61	19.23adv 1584	. . . . . . . . 9 |- (X e. C -> (E.t t e. B -> (F:Y-->X -> (y = X -> (y C_ A -> E.x e. B (F"x) C_ A)))))
63	62	3imp 1061	. . . . . . . 8 |- ((X e. C /\ E.t t e. B /\ F:Y-->X) -> (y = X -> (y C_ A -> E.x e. B (F"x) C_ A)))
64		fbasne0 10262	. . . . . . . . 9 |- (B e. fBas -> B =/= (/))
65		n0 2884	. . . . . . . . 9 |- (B =/= (/) <-> E.t t e. B)
66	64, 65	sylib 215	. . . . . . . 8 |- (B e. fBas -> E.t t e. B)
67	63, 66	syl3an2 1131	. . . . . . 7 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (y = X -> (y C_ A -> E.x e. B (F"x) C_ A)))
68	49, 67	jaod 469	. . . . . 6 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> ((E.t e. B y = (F"t) \/ y = X) -> (y C_ A -> E.x e. B (F"x) C_ A)))
69		elun 2741	. . . . . . 7 |- (y e. ({w | E.t e. B w = (F"t)} u. {X}) <-> (y e. {w | E.t e. B w = (F"t)} \/ y e. {X}))
70		visset 2295	. . . . . . . . 9 |- y e. _V
71		eqeq1 1890	. . . . . . . . . 10 |- (w = y -> (w = (F"t) <-> y = (F"t)))
72	71	rexbidv 2124	. . . . . . . . 9 |- (w = y -> (E.t e. B w = (F"t) <-> E.t e. B y = (F"t)))
73	70, 72	elab 2403	. . . . . . . 8 |- (y e. {w | E.t e. B w = (F"t)} <-> E.t e. B y = (F"t))
74		elsn 3058	. . . . . . . 8 |- (y e. {X} <-> y = X)
75	73, 74	orbi12i 277	. . . . . . 7 |- ((y e. {w | E.t e. B w = (F"t)} \/ y e. {X}) <-> (E.t e. B y = (F"t) \/ y = X))
76	69, 75	bitri 190	. . . . . 6 |- (y e. ({w | E.t e. B w = (F"t)} u. {X}) <-> (E.t e. B y = (F"t) \/ y = X))
77	68, 76	syl5ib 223	. . . . 5 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (y e. ({w | E.t e. B w = (F"t)} u. {X}) -> (y C_ A -> E.x e. B (F"x) C_ A)))
78	77	r19.23adv 2215	. . . 4 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A -> E.x e. B (F"x) C_ A))
79		ssun1 2767	. . . . . . . . 9 |- {w | E.t e. B w = (F"t)} C_ ({w | E.t e. B w = (F"t)} u. {X})
80	79	a1i 8	. . . . . . . 8 |- (((X e. C /\ B e. fBas /\ F:Y-->X) /\ (x e. B /\ (F"x) C_ A)) -> {w | E.t e. B w = (F"t)} C_ ({w | E.t e. B w = (F"t)} u. {X}))
81		eqid 1884	. . . . . . . . . . 11 |- (F"x) = (F"x)
82		imaeq2 4260	. . . . . . . . . . . . 13 |- (t = x -> (F"t) = (F"x))
83	82	eqeq2d 1895	. . . . . . . . . . . 12 |- (t = x -> ((F"x) = (F"t) <-> (F"x) = (F"x)))
84	83	rcla4ev 2381	. . . . . . . . . . 11 |- ((x e. B /\ (F"x) = (F"x)) -> E.t e. B (F"x) = (F"t))
85	81, 84	mpan2 760	. . . . . . . . . 10 |- (x e. B -> E.t e. B (F"x) = (F"t))
86	85	ad2antrl 442	. . . . . . . . 9 |- (((X e. C /\ B e. fBas /\ F:Y-->X) /\ (x e. B /\ (F"x) C_ A)) -> E.t e. B (F"x) = (F"t))
87		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
88	87	funimaex 4496	. . . . . . . . . . . 12 |- (Fun F -> (F"x) e. _V)
89		eqeq1 1890	. . . . . . . . . . . . . 14 |- (w = (F"x) -> (w = (F"t) <-> (F"x) = (F"t)))
90	89	rexbidv 2124	. . . . . . . . . . . . 13 |- (w = (F"x) -> (E.t e. B w = (F"t) <-> E.t e. B (F"x) = (F"t)))
91	90	elabg 2405	. . . . . . . . . . . 12 |- ((F"x) e. _V -> ((F"x) e. {w | E.t e. B w = (F"t)} <-> E.t e. B (F"x) = (F"t)))
92	9, 88, 91	3syl 24	. . . . . . . . . . 11 |- (F:Y-->X -> ((F"x) e. {w | E.t e. B w = (F"t)} <-> E.t e. B (F"x) = (F"t)))
93	92	3ad2ant3 899	. . . . . . . . . 10 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> ((F"x) e. {w | E.t e. B w = (F"t)} <-> E.t e. B (F"x) = (F"t)))
94	93	adantr 425	. . . . . . . . 9 |- (((X e. C /\ B e. fBas /\ F:Y-->X) /\ (x e. B /\ (F"x) C_ A)) -> ((F"x) e. {w | E.t e. B w = (F"t)} <-> E.t e. B (F"x) = (F"t)))
95	86, 94	mpbird 213	. . . . . . . 8 |- (((X e. C /\ B e. fBas /\ F:Y-->X) /\ (x e. B /\ (F"x) C_ A)) -> (F"x) e. {w | E.t e. B w = (F"t)})
96	80, 95	sseldd 2620	. . . . . . 7 |- (((X e. C /\ B e. fBas /\ F:Y-->X) /\ (x e. B /\ (F"x) C_ A)) -> (F"x) e. ({w | E.t e. B w = (F"t)} u. {X}))
97		simprr 451	. . . . . . 7 |- (((X e. C /\ B e. fBas /\ F:Y-->X) /\ (x e. B /\ (F"x) C_ A)) -> (F"x) C_ A)
98		sseq1 2637	. . . . . . . 8 |- (y = (F"x) -> (y C_ A <-> (F"x) C_ A))
99	98	rcla4ev 2381	. . . . . . 7 |- (((F"x) e. ({w | E.t e. B w = (F"t)} u. {X}) /\ (F"x) C_ A) -> E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A)
100	96, 97, 99	syl11anc 524	. . . . . 6 |- (((X e. C /\ B e. fBas /\ F:Y-->X) /\ (x e. B /\ (F"x) C_ A)) -> E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A)
101	100	exp32 408	. . . . 5 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (x e. B -> ((F"x) C_ A -> E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A)))
102	101	r19.23adv 2215	. . . 4 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (E.x e. B (F"x) C_ A -> E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A))
103	78, 102	impbid 574	. . 3 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A <-> E.x e. B (F"x) C_ A))
104	37, 103	anbi12d 690	. 2 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> ((A C_ U.({w | E.t e. B w = (F"t)} u. {X}) /\ E.y e. ({w | E.t e. B w = (F"t)} u. {X})y C_ A) <-> (A C_ X /\ E.x e. B (F"x) C_ A)))
105	3, 33, 104	3bitrd 603	1 |- ((X e. C /\ B e. fBas /\ F:Y-->X) -> (A e. ((X FilMap B)` F) <-> (A C_ X /\ E.x e. B (F"x) C_ A)))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  (/)c0 2875  {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:  elfilmap2 10313  fbfgfmeq 10315  flimfnei 10319  isflimf 10323  rnelfm 15593  fmfnfmlem1 15594  fmfnfm 15598  isfclusf 15625  filnet 15645

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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