Theorem hmphsyma 10528

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 1813	. . . 4 |- f e. V
2		eqid 1475	. . . . . 6 |- U.J = U.J
3		eqid 1475	. . . . . 6 |- U.K = U.K
4	2, 3	ishomeo 10517	. . . . 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		eleq1 1534	. . . . . . . 8 |- (g = `'f -> (g e. (K Homeo J) <-> `'f e. (K Homeo J)))
6	5	cla4egv 1863	. . . . . . 7 |- (`'f e. V -> (`'f e. (K Homeo J) -> E.g g e. (K Homeo J)))
7		cnvexg 3519	. . . . . . . . 9 |- (f e. V -> `'f e. V)
8	7	3ad2ant3 802	. . . . . . . 8 |- ((J e. Top /\ K e. Top /\ f e. V) -> `'f e. V)
9	8	adantr 389	. . . . . . 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)
10		f1orel 3692	. . . . . . . . . . . 12 |- (f:U.J-1-1-onto->U.K -> Rel f)
11	10	3ad2ant1 800	. . . . . . . . . . 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)
12	11	adantl 388	. . . . . . . . . 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)
13		f1ocnv 3701	. . . . . . . . . . . . . 14 |- (f:U.J-1-1-onto->U.K -> `'f:U.K-1-1-onto->U.J)
14	13	a1i 8	. . . . . . . . . . . . 13 |- ((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))
15		dfrel2 3485	. . . . . . . . . . . . . . . . . . . 20 |- (Rel f <-> `'`'f = f)
16	15	biimp 151	. . . . . . . . . . . . . . . . . . 19 |- (Rel f -> `'`'f = f)
17	16	eqcomd 1480	. . . . . . . . . . . . . . . . . 18 |- (Rel f -> f = `'`'f)
18	17	imaeq1d 3403	. . . . . . . . . . . . . . . . 17 |- (Rel f -> (f"x) = (`'`'f"x))
19	18	eleq1d 1540	. . . . . . . . . . . . . . . 16 |- (Rel f -> ((f"x) e. K <-> (`'`'f"x) e. K))
20	19	biimpd 153	. . . . . . . . . . . . . . 15 |- (Rel f -> ((f"x) e. K -> (`'`'f"x) e. K))
21	20	ad2antrr 404	. . . . . . . . . . . . . 14 |- (((Rel f /\ (J e. Top /\ K e. Top /\ f e. V)) /\ x e. J) -> ((f"x) e. K -> (`'`'f"x) e. K))
22	21	r19.20dva 1709	. . . . . . . . . . . . 13 |- ((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))
23		idd 61	. . . . . . . . . . . . 13 |- ((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))
24	14, 22, 23	3anim123d 900	. . . . . . . . . . . 12 |- ((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)))
25	24	ex 373	. . . . . . . . . . 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))))
26	25	imp3a 361	. . . . . . . . . 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)))
27	12, 26	mpcom 49	. . . . . . . . 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))
28		3ancomb 783	. . . . . . . . 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))
29	27, 28	sylib 198	. . . . . . . 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))
30	3, 2	ishomeo 10517	. . . . . . . . . . 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)))
31	30, 7	syl3an3 861	. . . . . . . . . 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)))
32	31	3com12 837	. . . . . . . . 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)))
33	32	adantr 389	. . . . . . . 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)))
34	29, 33	mpbird 196	. . . . . . 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))
35	6, 9, 34	sylc 68	. . . . . 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))
36	35	ex 373	. . . . 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)))
37	4, 36	sylbid 203	. . . 4 |- ((J e. Top /\ K e. Top /\ f e. V) -> (f e. (J Homeo K) -> E.g g e. (K Homeo J)))
38	1, 37	mp3an3 905	. . 3 |- ((J e. Top /\ K e. Top) -> (f e. (J Homeo K) -> E.g g e. (K Homeo J)))
39	38	19.23adv 1214	. 2 |- ((J e. Top /\ K e. Top) -> (E.f f e. (J Homeo K) -> E.g g e. (K Homeo J)))
40		hmph 10524	. 2 |- ((J e. Top /\ K e. Top) -> (J ~= K <-> E.f f e. (J Homeo K)))
41		hmph 10524	. . 3 |- ((K e. Top /\ J e. Top) -> (K ~= J <-> E.g g e. (K Homeo J)))
42	41	ancoms 436	. 2 |- ((J e. Top /\ K e. Top) -> (K ~= J <-> E.g g e. (K Homeo J)))
43	39, 40, 42	3imtr4d 543	1 |- ((J e. Top /\ K e. Top) -> (J ~= K -> K ~= J))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  A.wral 1645  Vcvv 1811  U.cuni 2503   class class class wbr 2619  `'ccnv 3169  "cima 3173  Rel wrel 3175  -1-1-onto->wf1o 3181  (class class class)co 3963  Topctop 7588   Homeo chomeosm 10513   ~= chomeo 10514

This theorem is referenced by:  hmphsym 10529  hmpher 10536  homindlem3 10551

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  df-hmph 10523

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