Theorem hmeobc 10239

Description: A homeomorphism is a bicontinuous bijection. (Contributed by FL, 1-Sep-2008.)

Hypotheses

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

Assertion

Ref	Expression
hmeobc	|- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. (J Homeo K) <-> (F:X-1-1-onto->Y /\ F e. (J Cn K) /\ `'F e. (K Cn J))))

Proof of Theorem hmeobc

Step	Hyp	Ref	Expression
1		simp1 876	. . . . 5 |- ((F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J) -> F:X-1-1-onto->Y)
2	1	adantl 424	. . . 4 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> F:X-1-1-onto->Y)
3		f1of 4635	. . . . . . 7 |- (F:X-1-1-onto->Y -> F:X-->Y)
4	3	3ad2ant1 897	. . . . . 6 |- ((F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J) -> F:X-->Y)
5	4	adantl 424	. . . . 5 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> F:X-->Y)
6		simp3 878	. . . . . 6 |- ((F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J) -> A.x e. K (`'F"x) e. J)
7	6	adantl 424	. . . . 5 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> A.x e. K (`'F"x) e. J)
8	5, 7	jca 310	. . . 4 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> (F:X-->Y /\ A.x e. K (`'F"x) e. J))
9		f1ocnv 4651	. . . . . . . 8 |- (F:X-1-1-onto->Y -> `'F:Y-1-1-onto->X)
10		f1of 4635	. . . . . . . 8 |- (`'F:Y-1-1-onto->X -> `'F:Y-->X)
11	9, 10	syl 12	. . . . . . 7 |- (F:X-1-1-onto->Y -> `'F:Y-->X)
12	11	3ad2ant1 897	. . . . . 6 |- ((F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J) -> `'F:Y-->X)
13	12	adantl 424	. . . . 5 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> `'F:Y-->X)
14		imacnvcnv 4388	. . . . . . . . . . 11 |- (`'`'F"x) = (F"x)
15	14	eqcomi 1888	. . . . . . . . . 10 |- (F"x) = (`'`'F"x)
16	15	eleq1i 1960	. . . . . . . . 9 |- ((F"x) e. K <-> (`'`'F"x) e. K)
17	16	ralbii 2127	. . . . . . . 8 |- (A.x e. J (F"x) e. K <-> A.x e. J (`'`'F"x) e. K)
18	17	biimpi 168	. . . . . . 7 |- (A.x e. J (F"x) e. K -> A.x e. J (`'`'F"x) e. K)
19	18	3ad2ant2 898	. . . . . 6 |- ((F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J) -> A.x e. J (`'`'F"x) e. K)
20	19	adantl 424	. . . . 5 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> A.x e. J (`'`'F"x) e. K)
21	13, 20	jca 310	. . . 4 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K))
22	2, 8, 21	3jca 1050	. . 3 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)) -> (F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K)))
23		simp1 876	. . . . 5 |- ((F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K)) -> F:X-1-1-onto->Y)
24	23	adantl 424	. . . 4 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K))) -> F:X-1-1-onto->Y)
25		frel 4566	. . . . . . . . 9 |- (F:X-->Y -> Rel F)
26	14	a1i 8	. . . . . . . . . . . . . . 15 |- (Rel F -> (`'`'F"x) = (F"x))
27	26	eleq1d 1963	. . . . . . . . . . . . . 14 |- (Rel F -> ((`'`'F"x) e. K <-> (F"x) e. K))
28	27	ralbidv 2123	. . . . . . . . . . . . 13 |- (Rel F -> (A.x e. J (`'`'F"x) e. K <-> A.x e. J (F"x) e. K))
29	28	biimpcd 172	. . . . . . . . . . . 12 |- (A.x e. J (`'`'F"x) e. K -> (Rel F -> A.x e. J (F"x) e. K))
30	29	adantl 424	. . . . . . . . . . 11 |- ((`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K) -> (Rel F -> A.x e. J (F"x) e. K))
31	30	com12 14	. . . . . . . . . 10 |- (Rel F -> ((`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K) -> A.x e. J (F"x) e. K))
32	31	a1d 15	. . . . . . . . 9 |- (Rel F -> (A.x e. K (`'F"x) e. J -> ((`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K) -> A.x e. J (F"x) e. K)))
33	25, 32	syl 12	. . . . . . . 8 |- (F:X-->Y -> (A.x e. K (`'F"x) e. J -> ((`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K) -> A.x e. J (F"x) e. K)))
34	33	imp 377	. . . . . . 7 |- ((F:X-->Y /\ A.x e. K (`'F"x) e. J) -> ((`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K) -> A.x e. J (F"x) e. K))
35	34	a1i 8	. . . . . 6 |- (F:X-1-1-onto->Y -> ((F:X-->Y /\ A.x e. K (`'F"x) e. J) -> ((`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K) -> A.x e. J (F"x) e. K)))
36	35	3imp 1061	. . . . 5 |- ((F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K)) -> A.x e. J (F"x) e. K)
37	36	adantl 424	. . . 4 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K))) -> A.x e. J (F"x) e. K)
38		simpr2r 936	. . . 4 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K))) -> A.x e. K (`'F"x) e. J)
39	24, 37, 38	3jca 1050	. . 3 |- (((J e. Top /\ K e. Top /\ F e. A) /\ (F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K))) -> (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J))
40	22, 39	impbida 577	. 2 |- ((J e. Top /\ K e. Top /\ F e. A) -> ((F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J) <-> (F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K))))
41		hmeobc.1	. . 3 |- X = U.J
42		hmeobc.2	. . 3 |- Y = U.K
43	41, 42	ishomeo 10235	. 2 |- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. (J Homeo K) <-> (F:X-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)))
44	41, 42	iscn 9034	. . . 4 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.x e. K (`'F"x) e. J)))
45	44	3adant3 896	. . 3 |- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.x e. K (`'F"x) e. J)))
46	42, 41	iscn 9034	. . . . 5 |- ((K e. Top /\ J e. Top) -> (`'F e. (K Cn J) <-> (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K)))
47	46	ancoms 484	. . . 4 |- ((J e. Top /\ K e. Top) -> (`'F e. (K Cn J) <-> (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K)))
48	47	3adant3 896	. . 3 |- ((J e. Top /\ K e. Top /\ F e. A) -> (`'F e. (K Cn J) <-> (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K)))
49	45, 48	3anbi23d 1171	. 2 |- ((J e. Top /\ K e. Top /\ F e. A) -> ((F:X-1-1-onto->Y /\ F e. (J Cn K) /\ `'F e. (K Cn J)) <-> (F:X-1-1-onto->Y /\ (F:X-->Y /\ A.x e. K (`'F"x) e. J) /\ (`'F:Y-->X /\ A.x e. J (`'`'F"x) e. K))))
50	40, 43, 49	3bitr4d 609	1 |- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. (J Homeo K) <-> (F:X-1-1-onto->Y /\ F e. (J Cn K) /\ `'F e. (K Cn J))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  U.cuni 3177  `'ccnv 3985  "cima 3989  Rel wrel 3991  -->wf 3994  -1-1-onto->wf1o 3997  (class class class)co 4884  Topctop 8857   Cn ccn 9028   Homeo chomeosm 10230

This theorem is referenced by:  comptoppr 10332  trhom 14983  hmeoclda 15421  conntoppr 15445  hmeocnv 15898

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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-cn 9030  df-homeo 10232

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