Theorem cnvhmpha 10511

Description: The converse of a homeomorphism is a homeomorphism.

Assertion

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

Proof of Theorem cnvhmpha

Step	Hyp	Ref	Expression
1		cnvexg 3525	. . 3 |- (F e. (J Homeo K) -> `'F e. V)
2		eqid 1478	. . . . . . . . . . . . . 14 |- U.J = U.J
3		eqid 1478	. . . . . . . . . . . . . 14 |- U.K = U.K
4	2, 3	hmeomap 10504	. . . . . . . . . . . . 13 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> F:U.J-1-1-onto->U.K))
5		f1ocnv 3707	. . . . . . . . . . . . 13 |- (F:U.J-1-1-onto->U.K -> `'F:U.K-1-1-onto->U.J)
6	4, 5	syl6 22	. . . . . . . . . . . 12 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> `'F:U.K-1-1-onto->U.J))
7	6	imp 350	. . . . . . . . . . 11 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> `'F:U.K-1-1-onto->U.J)
8		hmeocna 10505	. . . . . . . . . . . 12 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> A.x e. K (`'F"x) e. J))
9	8	imp 350	. . . . . . . . . . 11 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> A.x e. K (`'F"x) e. J)
10		hmeocnb 10506	. . . . . . . . . . . . 13 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> A.x e. J (F"x) e. K))
11	10	imp 350	. . . . . . . . . . . 12 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> A.x e. J (F"x) e. K)
12		f1orel 3698	. . . . . . . . . . . . . . . . . . 19 |- (F:U.J-1-1-onto->U.K -> Rel F)
13		dfrel2 3491	. . . . . . . . . . . . . . . . . . 19 |- (Rel F <-> `'`'F = F)
14	12, 13	sylib 198	. . . . . . . . . . . . . . . . . 18 |- (F:U.J-1-1-onto->U.K -> `'`'F = F)
15	4, 14	syl6 22	. . . . . . . . . . . . . . . . 17 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> `'`'F = F))
16	15	imp 350	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> `'`'F = F)
17	16	adantr 391	. . . . . . . . . . . . . . 15 |- ((((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) /\ x e. J) -> `'`'F = F)
18	17	imaeq1d 3409	. . . . . . . . . . . . . 14 |- ((((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) /\ x e. J) -> (`'`'F"x) = (F"x))
19	18	eleq1d 1543	. . . . . . . . . . . . 13 |- ((((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) /\ x e. J) -> ((`'`'F"x) e. K <-> (F"x) e. K))
20	19	ralbidva 1662	. . . . . . . . . . . 12 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> (A.x e. J (`'`'F"x) e. K <-> A.x e. J (F"x) e. K))
21	11, 20	mpbird 196	. . . . . . . . . . 11 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> A.x e. J (`'`'F"x) e. K)
22	7, 9, 21	3jca 821	. . . . . . . . . 10 |- (((J e. Top /\ K e. Top) /\ 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	ex 373	. . . . . . . . 9 |- ((J e. Top /\ K e. Top) -> (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 438	. . . . . . . 8 |- ((K e. Top /\ J e. Top) -> (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 801	. . . . . . 7 |- ((K e. Top /\ J e. Top /\ `'F e. V) -> (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	3, 2	ishomeo 10503	. . . . . . 7 |- ((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)))
27	25, 26	sylibrd 204	. . . . . 6 |- ((K e. Top /\ J e. Top /\ `'F e. V) -> (F e. (J Homeo K) -> `'F e. (K Homeo J)))
28	27	3exp 834	. . . . 5 |- (K e. Top -> (J e. Top -> (`'F e. V -> (F e. (J Homeo K) -> `'F e. (K Homeo J)))))
29	28	impcom 351	. . . 4 |- ((J e. Top /\ K e. Top) -> (`'F e. V -> (F e. (J Homeo K) -> `'F e. (K Homeo J))))
30	29	com3l 34	. . 3 |- (`'F e. V -> (F e. (J Homeo K) -> ((J e. Top /\ K e. Top) -> `'F e. (K Homeo J))))
31	1, 30	mpcom 49	. 2 |- (F e. (J Homeo K) -> ((J e. Top /\ K e. Top) -> `'F e. (K Homeo J)))
32	31	com12 11	1 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> `'F e. (K Homeo J)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814  U.cuni 2507  `'ccnv 3175  "cima 3179  Rel wrel 3181  -1-1-onto->wf1o 3187  (class class class)co 3969  Topctop 7590   Homeo chomeosm 10499

This theorem is referenced by:  hmeogrp 10524

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-opr 3971  df-oprab 3972  df-homeo 10501

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