Theorem cnvhmphb 10526

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 3519	. . 3 |- (`'F e. (J Homeo K) -> `'`'F e. V)
2		dfrel2 3485	. . . . . . . . . 10 |- (Rel F <-> `'`'F = F)
3	2	biimp 151	. . . . . . . . 9 |- (Rel F -> `'`'F = F)
4	3	eleq1d 1540	. . . . . . . 8 |- (Rel F -> (`'`'F e. V <-> F e. V))
5		eqid 1475	. . . . . . . . . . . . . . . . . . . 20 |- U.J = U.J
6		eqid 1475	. . . . . . . . . . . . . . . . . . . 20 |- U.K = U.K
7	5, 6	hmeomap 10518	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> `'F:U.J-1-1-onto->U.K))
8		f1ocnvb 3702	. . . . . . . . . . . . . . . . . . . 20 |- (Rel F -> (F:U.K-1-1-onto->U.J <-> `'F:U.J-1-1-onto->U.K))
9	8	biimprd 154	. . . . . . . . . . . . . . . . . . 19 |- (Rel F -> (`'F:U.J-1-1-onto->U.K -> F:U.K-1-1-onto->U.J))
10	7, 9	syl9 57	. . . . . . . . . . . . . . . . . 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 362	. . . . . . . . . . . . . . . . 17 |- ((((J e. Top /\ K e. Top) /\ Rel F) /\ `'F e. (J Homeo K)) -> F:U.K-1-1-onto->U.J)
12		hmeocna 10519	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> A.x e. K (`'`'F"x) e. J))
13	3	imaeq1d 3403	. . . . . . . . . . . . . . . . . . . . . 22 |- (Rel F -> (`'`'F"x) = (F"x))
14	13	eleq1d 1540	. . . . . . . . . . . . . . . . . . . . 21 |- (Rel F -> ((`'`'F"x) e. J <-> (F"x) e. J))
15	14	ralbidv 1663	. . . . . . . . . . . . . . . . . . . 20 |- (Rel F -> (A.x e. K (`'`'F"x) e. J <-> A.x e. K (F"x) e. J))
16	15	biimpd 153	. . . . . . . . . . . . . . . . . . 19 |- (Rel F -> (A.x e. K (`'`'F"x) e. J -> A.x e. K (F"x) e. J))
17	12, 16	syl9 57	. . . . . . . . . . . . . . . . . 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 362	. . . . . . . . . . . . . . . . 17 |- ((((J e. Top /\ K e. Top) /\ Rel F) /\ `'F e. (J Homeo K)) -> A.x e. K (F"x) e. J)
19		hmeocnb 10520	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ K e. Top) -> (`'F e. (J Homeo K) -> A.x e. J (`'F"x) e. K))
20	19	adantr 389	. . . . . . . . . . . . . . . . . 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 350	. . . . . . . . . . . . . . . . 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 819	. . . . . . . . . . . . . . . 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 376	. . . . . . . . . . . . . . 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 436	. . . . . . . . . . . . . 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 799	. . . . . . . . . . . . 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 350	. . . . . . . . . . . 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 10517	. . . . . . . . . . . . 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 389	. . . . . . . . . . . 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 204	. . . . . . . . . . 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 849	. . . . . . . . . 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 351	. . . . . . . . 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 34	. . . . . . . 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 214	. . . . . . 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 66	. . . . . 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 35	. . . . 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 830	. . . 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 34	. . 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 49	. 2 |- (`'F e. (J Homeo K) -> ((J e. Top /\ K e. Top /\ Rel F) -> F e. (K Homeo J)))
39	38	com12 11	1 |- ((J e. Top /\ K e. Top /\ Rel F) -> (`'F e. (J Homeo K) -> F e. (K Homeo J)))

Syntax hints:   -> wi 3   <-> wb 146   /\ 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 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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