Theorem cnvhmpha 10525

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 3519	. . 3 |- (F e. (J Homeo K) -> `'F e. V)
2		eqid 1475	. . . . . . . . . . . . . 14 |- U.J = U.J
3		eqid 1475	. . . . . . . . . . . . . 14 |- U.K = U.K
4	2, 3	hmeomap 10518	. . . . . . . . . . . . 13 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> F:U.J-1-1-onto->U.K))
5		f1ocnv 3701	. . . . . . . . . . . . 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 10519	. . . . . . . . . . . 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 10520	. . . . . . . . . . . . 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 3692	. . . . . . . . . . . . . . . . . . 19 |- (F:U.J-1-1-onto->U.K -> Rel F)
13		dfrel2 3485	. . . . . . . . . . . . . . . . . . 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 389	. . . . . . . . . . . . . . 15 |- ((((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) /\ x e. J) -> `'`'F = F)
18	17	imaeq1d 3403	. . . . . . . . . . . . . 14 |- ((((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) /\ x e. J) -> (`'`'F"x) = (F"x))
19	18	eleq1d 1540	. . . . . . . . . . . . 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 1659	. . . . . . . . . . . 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 819	. . . . . . . . . 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 436	. . . . . . . 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 799	. . . . . . 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 10517	. . . . . . 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 832	. . . . 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 775   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  U.cuni 2503  `'ccnv 3169  "cima 3173  Rel wrel 3175  -1-1-onto->wf1o 3181  (class class class)co 3963  Topctop 7588   Homeo chomeosm 10513

This theorem is referenced by:  hmeogrp 10538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-homeo 10515

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