Theorem cnvhmphb 14880

Description: The converse of a homeomorphism is a homeomorphism.

Assertion

Ref	Expression
cnvhmphb	|- ((J e. Top /\ K e. Top /\ Rel F) -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))

Proof of Theorem cnvhmphb

Step	Hyp	Ref	Expression
1		cnvexg 4424	. . 3 |- (`'F e. (J Homeo K) -> `'`'F e. _V)
2		dfrel2 4358	. . . . . . . . . 10 |- (Rel F <-> `'`'F = F)
3	2	biimpi 168	. . . . . . . . 9 |- (Rel F -> `'`'F = F)
4	3	eleq1d 1963	. . . . . . . 8 |- (Rel F -> (`'`'F e. _V <-> F e. _V))
5		eqid 1884	. . . . . . . . . . . . . . . . . . . 20 |- U.J = U.J
6		eqid 1884	. . . . . . . . . . . . . . . . . . . 20 |- U.K = U.K
7	5, 6	hmeomap 10236	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> `'F:U.J-1-1-onto->U.K))
8		f1ocnvb 4653	. . . . . . . . . . . . . . . . . . . 20 |- (Rel F -> (F:U.K-1-1-onto->U.J <-> `'F:U.J-1-1-onto->U.K))
9	8	biimprd 171	. . . . . . . . . . . . . . . . . . 19 |- (Rel F -> (`'F:U.J-1-1-onto->U.K -> F:U.K-1-1-onto->U.J))
10	7, 9	syl9 71	. . . . . . . . . . . . . . . . . 18 |- ((J e. Top /\ K e. Top) -> (Rel F -> (`'F e. (J Homeo K) -> F:U.K-1-1-onto->U.J)))
11	10	imp31 389	. . . . . . . . . . . . . . . . 17 |- ((((J e. Top /\ K e. Top) /\ Rel F) /\ `'F e. (J Homeo K)) -> F:U.K-1-1-onto->U.J)
12		hmeocna 10237	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> A.x e. K (`'`'F"x) e. J))
13	3	imaeq1d 4263	. . . . . . . . . . . . . . . . . . . . . 22 |- (Rel F -> (`'`'F"x) = (F"x))
14	13	eleq1d 1963	. . . . . . . . . . . . . . . . . . . . 21 |- (Rel F -> ((`'`'F"x) e. J <-> (F"x) e. J))
15	14	ralbidv 2123	. . . . . . . . . . . . . . . . . . . 20 |- (Rel F -> (A.x e. K (`'`'F"x) e. J <-> A.x e. K (F"x) e. J))
16	15	biimpd 170	. . . . . . . . . . . . . . . . . . 19 |- (Rel F -> (A.x e. K (`'`'F"x) e. J -> A.x e. K (F"x) e. J))
17	12, 16	syl9 71	. . . . . . . . . . . . . . . . . 18 |- ((J e. Top /\ K e. Top) -> (Rel F -> (`'F e. (J Homeo K) -> A.x e. K (F"x) e. J)))
18	17	imp31 389	. . . . . . . . . . . . . . . . 17 |- ((((J e. Top /\ K e. Top) /\ Rel F) /\ `'F e. (J Homeo K)) -> A.x e. K (F"x) e. J)
19		hmeocnb 10238	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> A.x e. J (`'F"x) e. K))
20	19	adantr 425	. . . . . . . . . . . . . . . . . 18 |- (((J e. Top /\ K e. Top) /\ Rel F) -> (`'F e. (J Homeo K) -> A.x e. J (`'F"x) e. K))
21	20	imp 377	. . . . . . . . . . . . . . . . 17 |- ((((J e. Top /\ K e. Top) /\ Rel F) /\ `'F e. (J Homeo K)) -> A.x e. J (`'F"x) e. K)
22	11, 18, 21	3jca 1050	. . . . . . . . . . . . . . . 16 |- ((((J e. Top /\ K e. Top) /\ Rel F) /\ `'F e. (J Homeo K)) -> (F:U.K-1-1-onto->U.J /\ A.x e. K (F"x) e. J /\ A.x e. J (`'F"x) e. K))
23	22	exp31 407	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ K e. Top) -> (Rel F -> (`'F e. (J Homeo K) -> (F:U.K-1-1-onto->U.J /\ A.x e. K (F"x) e. J /\ A.x e. J (`'F"x) e. K))))
24	23	ancoms 484	. . . . . . . . . . . . . 14 |- ((K e. Top /\ J e. Top) -> (Rel F -> (`'F e. (J Homeo K) -> (F:U.K-1-1-onto->U.J /\ A.x e. K (F"x) e. J /\ A.x e. J (`'F"x) e. K))))
25	24	3adant3 896	. . . . . . . . . . . . 13 |- ((K e. Top /\ J e. Top /\ F e. _V) -> (Rel F -> (`'F e. (J Homeo K) -> (F:U.K-1-1-onto->U.J /\ A.x e. K (F"x) e. J /\ A.x e. J (`'F"x) e. K))))
26	25	imp 377	. . . . . . . . . . . 12 |- (((K e. Top /\ J e. Top /\ F e. _V) /\ Rel F) -> (`'F e. (J Homeo K) -> (F:U.K-1-1-onto->U.J /\ A.x e. K (F"x) e. J /\ A.x e. J (`'F"x) e. K)))
27	6, 5	ishomeo 10235	. . . . . . . . . . . . 13 |- ((K e. Top /\ J e. Top /\ F e. _V) -> (F e. (K Homeo J) <-> (F:U.K-1-1-onto->U.J /\ A.x e. K (F"x) e. J /\ A.x e. J (`'F"x) e. K)))
28	27	adantr 425	. . . . . . . . . . . 12 |- (((K e. Top /\ J e. Top /\ F e. _V) /\ Rel F) -> (F e. (K Homeo J) <-> (F:U.K-1-1-onto->U.J /\ A.x e. K (F"x) e. J /\ A.x e. J (`'F"x) e. K)))
29	26, 28	sylibrd 221	. . . . . . . . . . 11 |- (((K e. Top /\ J e. Top /\ F e. _V) /\ Rel F) -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))
30	29	3exp1 1084	. . . . . . . . . 10 |- (K e. Top -> (J e. Top -> (F e. _V -> (Rel F -> (`'F e. (J Homeo K) -> F e. (K Homeo J))))))
31	30	impcom 378	. . . . . . . . 9 |- ((J e. Top /\ K e. Top) -> (F e. _V -> (Rel F -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))))
32	31	com3l 38	. . . . . . . 8 |- (F e. _V -> (Rel F -> ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))))
33	4, 32	syl6bi 231	. . . . . . 7 |- (Rel F -> (`'`'F e. _V -> (Rel F -> ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> F e. (K Homeo J))))))
34	33	pm2.43a 80	. . . . . 6 |- (Rel F -> (`'`'F e. _V -> ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))))
35	34	com3r 39	. . . . 5 |- ((J e. Top /\ K e. Top) -> (Rel F -> (`'`'F e. _V -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))))
36	35	3impia 1064	. . . 4 |- ((J e. Top /\ K e. Top /\ Rel F) -> (`'`'F e. _V -> (`'F e. (J Homeo K) -> F e. (K Homeo J))))
37	36	com3l 38	. . 3 |- (`'`'F e. _V -> (`'F e. (J Homeo K) -> ((J e. Top /\ K e. Top /\ Rel F) -> F e. (K Homeo J))))
38	1, 37	mpcom 60	. 2 |- (`'F e. (J Homeo K) -> ((J e. Top /\ K e. Top /\ Rel F) -> F e. (K Homeo J)))
39	38	com12 14	1 |- ((J e. Top /\ K e. Top /\ Rel F) -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  U.cuni 3177  `'ccnv 3985  "cima 3989  Rel wrel 3991  -1-1-onto->wf1o 3997  (class class class)co 4884  Topctop 8857   Homeo chomeosm 10230

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-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-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-homeo 10232

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