Theorem hmphsyma 14882

Description: "Is homeomorph to" is symmetric.

Assertion

Ref	Expression
hmphsyma	|- ((J e. Top /\ K e. Top) -> (J ~= K -> K ~= J))

Proof of Theorem hmphsyma

Step	Hyp	Ref	Expression
1		visset 2295	. . . 4 |- f e. _V
2		eqid 1884	. . . . . 6 |- U.J = U.J
3		eqid 1884	. . . . . 6 |- U.K = U.K
4	2, 3	ishomeo 10235	. . . . 5 |- ((J e. Top /\ K e. Top /\ f e. _V) -> (f e. (J Homeo K) <-> (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)))
5		cnvexg 4424	. . . . . . . . 9 |- (f e. _V -> `'f e. _V)
6	5	3ad2ant3 899	. . . . . . . 8 |- ((J e. Top /\ K e. Top /\ f e. _V) -> `'f e. _V)
7	6	adantr 425	. . . . . . 7 |- (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)) -> `'f e. _V)
8		f1orel 4638	. . . . . . . . . . . 12 |- (f:U.J-1-1-onto->U.K -> Rel f)
9	8	3ad2ant1 897	. . . . . . . . . . 11 |- ((f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J) -> Rel f)
10	9	adantl 424	. . . . . . . . . 10 |- (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)) -> Rel f)
11		f1ocnv 4651	. . . . . . . . . . . . 13 |- (f:U.J-1-1-onto->U.K -> `'f:U.K-1-1-onto->U.J)
12	11	a1i 8	. . . . . . . . . . . 12 |- ((Rel f /\ (J e. Top /\ K e. Top /\ f e. _V)) -> (f:U.J-1-1-onto->U.K -> `'f:U.K-1-1-onto->U.J))
13		dfrel2 4358	. . . . . . . . . . . . . . . . . . 19 |- (Rel f <-> `'`'f = f)
14	13	biimpi 168	. . . . . . . . . . . . . . . . . 18 |- (Rel f -> `'`'f = f)
15	14	eqcomd 1889	. . . . . . . . . . . . . . . . 17 |- (Rel f -> f = `'`'f)
16	15	imaeq1d 4263	. . . . . . . . . . . . . . . 16 |- (Rel f -> (f"x) = (`'`'f"x))
17	16	eleq1d 1963	. . . . . . . . . . . . . . 15 |- (Rel f -> ((f"x) e. K <-> (`'`'f"x) e. K))
18	17	biimpd 170	. . . . . . . . . . . . . 14 |- (Rel f -> ((f"x) e. K -> (`'`'f"x) e. K))
19	18	ad2antrr 440	. . . . . . . . . . . . 13 |- (((Rel f /\ (J e. Top /\ K e. Top /\ f e. _V)) /\ x e. J) -> ((f"x) e. K -> (`'`'f"x) e. K))
20	19	ralimdvaa 2171	. . . . . . . . . . . 12 |- ((Rel f /\ (J e. Top /\ K e. Top /\ f e. _V)) -> (A.x e. J (f"x) e. K -> A.x e. J (`'`'f"x) e. K))
21		idd 75	. . . . . . . . . . . 12 |- ((Rel f /\ (J e. Top /\ K e. Top /\ f e. _V)) -> (A.x e. K (`'f"x) e. J -> A.x e. K (`'f"x) e. J))
22	12, 20, 21	3anim123d 1175	. . . . . . . . . . 11 |- ((Rel f /\ (J e. Top /\ K e. Top /\ f e. _V)) -> ((f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J) -> (`'f:U.K-1-1-onto->U.J /\ A.x e. J (`'`'f"x) e. K /\ A.x e. K (`'f"x) e. J)))
23	22	expimpd 404	. . . . . . . . . 10 |- (Rel f -> (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)) -> (`'f:U.K-1-1-onto->U.J /\ A.x e. J (`'`'f"x) e. K /\ A.x e. K (`'f"x) e. J)))
24	10, 23	mpcom 60	. . . . . . . . 9 |- (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)) -> (`'f:U.K-1-1-onto->U.J /\ A.x e. J (`'`'f"x) e. K /\ A.x e. K (`'f"x) e. J))
25		3ancomb 866	. . . . . . . . 9 |- ((`'f:U.K-1-1-onto->U.J /\ A.x e. J (`'`'f"x) e. K /\ A.x e. K (`'f"x) e. 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))
26	24, 25	sylib 215	. . . . . . . 8 |- (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. 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	3, 2	ishomeo 10235	. . . . . . . . . . 11 |- ((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, 5	syl3an3 1132	. . . . . . . . . 10 |- ((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)))
29	28	3com12 1071	. . . . . . . . 9 |- ((J e. Top /\ K 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)))
30	29	adantr 425	. . . . . . . 8 |- (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)) -> (`'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)))
31	26, 30	mpbird 213	. . . . . . 7 |- (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)) -> `'f e. (K Homeo J))
32		eleq1 1957	. . . . . . . 8 |- (g = `'f -> (g e. (K Homeo J) <-> `'f e. (K Homeo J)))
33	32	cla4egv 2365	. . . . . . 7 |- (`'f e. _V -> (`'f e. (K Homeo J) -> E.g g e. (K Homeo J)))
34	7, 31, 33	sylc 83	. . . . . 6 |- (((J e. Top /\ K e. Top /\ f e. _V) /\ (f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)) -> E.g g e. (K Homeo J))
35	34	ex 402	. . . . 5 |- ((J e. Top /\ K e. Top /\ f e. _V) -> ((f:U.J-1-1-onto->U.K /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J) -> E.g g e. (K Homeo J)))
36	4, 35	sylbid 220	. . . 4 |- ((J e. Top /\ K e. Top /\ f e. _V) -> (f e. (J Homeo K) -> E.g g e. (K Homeo J)))
37	1, 36	mp3an3 1180	. . 3 |- ((J e. Top /\ K e. Top) -> (f e. (J Homeo K) -> E.g g e. (K Homeo J)))
38	37	19.23adv 1584	. 2 |- ((J e. Top /\ K e. Top) -> (E.f f e. (J Homeo K) -> E.g g e. (K Homeo J)))
39		hmph 10241	. 2 |- ((J e. Top /\ K e. Top) -> (J ~= K <-> E.f f e. (J Homeo K)))
40		hmph 10241	. . 3 |- ((K e. Top /\ J e. Top) -> (K ~= J <-> E.g g e. (K Homeo J)))
41	40	ancoms 484	. 2 |- ((J e. Top /\ K e. Top) -> (K ~= J <-> E.g g e. (K Homeo J)))
42	38, 39, 41	3imtr4d 602	1 |- ((J e. Top /\ K e. Top) -> (J ~= K -> K ~= J))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  _Vcvv 2292  U.cuni 3177   class class class wbr 3338  `'ccnv 3985  "cima 3989  Rel wrel 3991  -1-1-onto->wf1o 3997  (class class class)co 4884  Topctop 8857   Homeo chomeosm 10230   ~= chomeo 10231

This theorem is referenced by:  hmphsym 14883  hmpher 14890  homindlem3 14900

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  df-hmph 10233

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