Theorem iscncl 9047

Description: A definition of a continuous function using closed sets. Bourbaki TG I.9 Th. 1 (d). (Contributed by FL, 30-Jan-2007.)

Hypotheses

Ref	Expression
iscncl.1	|- X = U.J
iscncl.2	|- Y = U.K

Assertion

Ref	Expression
iscncl	|- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J))))

Distinct variable groups:   y,F   y,J   y,K   y,X   y,Y

Proof of Theorem iscncl

Step	Hyp	Ref	Expression
1		iscncl.1	. . 3 |- X = U.J
2		iscncl.2	. . 3 |- Y = U.K
3	1, 2	iscn 9034	. 2 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.x e. K (`'F"x) e. J)))
4		fimacnv 4783	. . . . . . . . . . . . . . . . 17 |- (F:X-->Y -> (`'F"Y) = X)
5	4	eqcomd 1889	. . . . . . . . . . . . . . . 16 |- (F:X-->Y -> X = (`'F"Y))
6	5	difeq1d 2725	. . . . . . . . . . . . . . 15 |- (F:X-->Y -> (X \ (`'F"y)) = ((`'F"Y) \ (`'F"y)))
7		ffun 4565	. . . . . . . . . . . . . . . 16 |- (F:X-->Y -> Fun F)
8		funcnvcnv 4473	. . . . . . . . . . . . . . . 16 |- (Fun F -> Fun `'`'F)
9		imadif 4493	. . . . . . . . . . . . . . . 16 |- (Fun `'`'F -> (`'F"(Y \ y)) = ((`'F"Y) \ (`'F"y)))
10	7, 8, 9	3syl 24	. . . . . . . . . . . . . . 15 |- (F:X-->Y -> (`'F"(Y \ y)) = ((`'F"Y) \ (`'F"y)))
11	6, 10	eqtr4d 1928	. . . . . . . . . . . . . 14 |- (F:X-->Y -> (X \ (`'F"y)) = (`'F"(Y \ y)))
12	11	a1i12 9	. . . . . . . . . . . . 13 |- (y e. (Clsd` K) -> (A.x e. K (`'F"x) e. J -> (F:X-->Y -> (X \ (`'F"y)) = (`'F"(Y \ y)))))
13	12	com13 37	. . . . . . . . . . . 12 |- (F:X-->Y -> (A.x e. K (`'F"x) e. J -> (y e. (Clsd` K) -> (X \ (`'F"y)) = (`'F"(Y \ y)))))
14	13	a1i 8	. . . . . . . . . . 11 |- (K e. Top -> (F:X-->Y -> (A.x e. K (`'F"x) e. J -> (y e. (Clsd` K) -> (X \ (`'F"y)) = (`'F"(Y \ y))))))
15	14	imp41 395	. . . . . . . . . 10 |- ((((K e. Top /\ F:X-->Y) /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> (X \ (`'F"y)) = (`'F"(Y \ y)))
16		simplr 449	. . . . . . . . . . . 12 |- (((K e. Top /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> A.x e. K (`'F"x) e. J)
17	2	cldopn 8948	. . . . . . . . . . . . 13 |- ((K e. Top /\ y e. (Clsd` K)) -> (Y \ y) e. K)
18	17	adantlr 429	. . . . . . . . . . . 12 |- (((K e. Top /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> (Y \ y) e. K)
19		imaeq2 4260	. . . . . . . . . . . . . 14 |- (x = (Y \ y) -> (`'F"x) = (`'F"(Y \ y)))
20	19	eleq1d 1963	. . . . . . . . . . . . 13 |- (x = (Y \ y) -> ((`'F"x) e. J <-> (`'F"(Y \ y)) e. J))
21	20	rcla4cva 2379	. . . . . . . . . . . 12 |- ((A.x e. K (`'F"x) e. J /\ (Y \ y) e. K) -> (`'F"(Y \ y)) e. J)
22	16, 18, 21	syl11anc 524	. . . . . . . . . . 11 |- (((K e. Top /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> (`'F"(Y \ y)) e. J)
23	22	adantllr 433	. . . . . . . . . 10 |- ((((K e. Top /\ F:X-->Y) /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> (`'F"(Y \ y)) e. J)
24	15, 23	eqeltrd 1971	. . . . . . . . 9 |- ((((K e. Top /\ F:X-->Y) /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> (X \ (`'F"y)) e. J)
25	24	exp41 413	. . . . . . . 8 |- (K e. Top -> (F:X-->Y -> (A.x e. K (`'F"x) e. J -> (y e. (Clsd` K) -> (X \ (`'F"y)) e. J))))
26	25	adantl 424	. . . . . . 7 |- ((J e. Top /\ K e. Top) -> (F:X-->Y -> (A.x e. K (`'F"x) e. J -> (y e. (Clsd` K) -> (X \ (`'F"y)) e. J))))
27	26	imp41 395	. . . . . 6 |- (((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> (X \ (`'F"y)) e. J)
28	1	iscld2 8946	. . . . . . . . 9 |- ((J e. Top /\ (`'F"y) C_ X) -> ((`'F"y) e. (Clsd` J) <-> (X \ (`'F"y)) e. J))
29		cnvimass 4286	. . . . . . . . . . 11 |- (`'F"y) C_ dom F
30	29	a1i 8	. . . . . . . . . 10 |- (F:X-->Y -> (`'F"y) C_ dom F)
31		fdm 4567	. . . . . . . . . 10 |- (F:X-->Y -> dom F = X)
32	30, 31	sseqtrd 2653	. . . . . . . . 9 |- (F:X-->Y -> (`'F"y) C_ X)
33	28, 32	sylan2 500	. . . . . . . 8 |- ((J e. Top /\ F:X-->Y) -> ((`'F"y) e. (Clsd` J) <-> (X \ (`'F"y)) e. J))
34	33	adantlr 429	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ F:X-->Y) -> ((`'F"y) e. (Clsd` J) <-> (X \ (`'F"y)) e. J))
35	34	ad2antrr 440	. . . . . 6 |- (((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> ((`'F"y) e. (Clsd` J) <-> (X \ (`'F"y)) e. J))
36	27, 35	mpbird 213	. . . . 5 |- (((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.x e. K (`'F"x) e. J) /\ y e. (Clsd` K)) -> (`'F"y) e. (Clsd` J))
37	36	r19.21aiva 2176	. . . 4 |- ((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.x e. K (`'F"x) e. J) -> A.y e. (Clsd` K)(`'F"y) e. (Clsd` J))
38	5	difeq1d 2725	. . . . . . . . . . . . . . 15 |- (F:X-->Y -> (X \ (`'F"x)) = ((`'F"Y) \ (`'F"x)))
39		imadif 4493	. . . . . . . . . . . . . . . 16 |- (Fun `'`'F -> (`'F"(Y \ x)) = ((`'F"Y) \ (`'F"x)))
40	7, 8, 39	3syl 24	. . . . . . . . . . . . . . 15 |- (F:X-->Y -> (`'F"(Y \ x)) = ((`'F"Y) \ (`'F"x)))
41	38, 40	eqtr4d 1928	. . . . . . . . . . . . . 14 |- (F:X-->Y -> (X \ (`'F"x)) = (`'F"(Y \ x)))
42	41	a1i12 9	. . . . . . . . . . . . 13 |- (x e. K -> (A.y e. (Clsd` K)(`'F"y) e. (Clsd` J) -> (F:X-->Y -> (X \ (`'F"x)) = (`'F"(Y \ x)))))
43	42	com13 37	. . . . . . . . . . . 12 |- (F:X-->Y -> (A.y e. (Clsd` K)(`'F"y) e. (Clsd` J) -> (x e. K -> (X \ (`'F"x)) = (`'F"(Y \ x)))))
44	43	a1i 8	. . . . . . . . . . 11 |- (K e. Top -> (F:X-->Y -> (A.y e. (Clsd` K)(`'F"y) e. (Clsd` J) -> (x e. K -> (X \ (`'F"x)) = (`'F"(Y \ x))))))
45	44	imp41 395	. . . . . . . . . 10 |- ((((K e. Top /\ F:X-->Y) /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> (X \ (`'F"x)) = (`'F"(Y \ x)))
46		simplr 449	. . . . . . . . . . . 12 |- (((K e. Top /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> A.y e. (Clsd` K)(`'F"y) e. (Clsd` J))
47	2	opncld 8950	. . . . . . . . . . . . 13 |- ((K e. Top /\ x e. K) -> (Y \ x) e. (Clsd` K))
48	47	adantlr 429	. . . . . . . . . . . 12 |- (((K e. Top /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> (Y \ x) e. (Clsd` K))
49		imaeq2 4260	. . . . . . . . . . . . . 14 |- (y = (Y \ x) -> (`'F"y) = (`'F"(Y \ x)))
50	49	eleq1d 1963	. . . . . . . . . . . . 13 |- (y = (Y \ x) -> ((`'F"y) e. (Clsd` J) <-> (`'F"(Y \ x)) e. (Clsd` J)))
51	50	rcla4cva 2379	. . . . . . . . . . . 12 |- ((A.y e. (Clsd` K)(`'F"y) e. (Clsd` J) /\ (Y \ x) e. (Clsd` K)) -> (`'F"(Y \ x)) e. (Clsd` J))
52	46, 48, 51	syl11anc 524	. . . . . . . . . . 11 |- (((K e. Top /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> (`'F"(Y \ x)) e. (Clsd` J))
53	52	adantllr 433	. . . . . . . . . 10 |- ((((K e. Top /\ F:X-->Y) /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> (`'F"(Y \ x)) e. (Clsd` J))
54	45, 53	eqeltrd 1971	. . . . . . . . 9 |- ((((K e. Top /\ F:X-->Y) /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> (X \ (`'F"x)) e. (Clsd` J))
55	54	exp41 413	. . . . . . . 8 |- (K e. Top -> (F:X-->Y -> (A.y e. (Clsd` K)(`'F"y) e. (Clsd` J) -> (x e. K -> (X \ (`'F"x)) e. (Clsd` J)))))
56	55	adantl 424	. . . . . . 7 |- ((J e. Top /\ K e. Top) -> (F:X-->Y -> (A.y e. (Clsd` K)(`'F"y) e. (Clsd` J) -> (x e. K -> (X \ (`'F"x)) e. (Clsd` J)))))
57	56	imp41 395	. . . . . 6 |- (((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> (X \ (`'F"x)) e. (Clsd` J))
58	1	isopn2 8949	. . . . . . . . 9 |- ((J e. Top /\ (`'F"x) C_ X) -> ((`'F"x) e. J <-> (X \ (`'F"x)) e. (Clsd` J)))
59		cnvimass 4286	. . . . . . . . . . 11 |- (`'F"x) C_ dom F
60	59	a1i 8	. . . . . . . . . 10 |- (F:X-->Y -> (`'F"x) C_ dom F)
61	60, 31	sseqtrd 2653	. . . . . . . . 9 |- (F:X-->Y -> (`'F"x) C_ X)
62	58, 61	sylan2 500	. . . . . . . 8 |- ((J e. Top /\ F:X-->Y) -> ((`'F"x) e. J <-> (X \ (`'F"x)) e. (Clsd` J)))
63	62	adantlr 429	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ F:X-->Y) -> ((`'F"x) e. J <-> (X \ (`'F"x)) e. (Clsd` J)))
64	63	ad2antrr 440	. . . . . 6 |- (((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> ((`'F"x) e. J <-> (X \ (`'F"x)) e. (Clsd` J)))
65	57, 64	mpbird 213	. . . . 5 |- (((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) /\ x e. K) -> (`'F"x) e. J)
66	65	r19.21aiva 2176	. . . 4 |- ((((J e. Top /\ K e. Top) /\ F:X-->Y) /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)) -> A.x e. K (`'F"x) e. J)
67	37, 66	impbida 577	. . 3 |- (((J e. Top /\ K e. Top) /\ F:X-->Y) -> (A.x e. K (`'F"x) e. J <-> A.y e. (Clsd` K)(`'F"y) e. (Clsd` J)))
68	67	pm5.32da 711	. 2 |- ((J e. Top /\ K e. Top) -> ((F:X-->Y /\ A.x e. K (`'F"x) e. J) <-> (F:X-->Y /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J))))
69	3, 68	bitrd 587	1 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   \ cdif 2590   C_ wss 2593  U.cuni 3177  `'ccnv 3985  dom cdm 3986  "cima 3989  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857  Clsdccld 8936   Cn ccn 9028

This theorem is referenced by:  cnclima 9048  cncls 15419  paste 15893

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-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-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-cld 8939  df-cn 9030

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