Theorem comptoppr 10332

Description: Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.)

Assertion

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

Proof of Theorem comptoppr

Step	Hyp	Ref	Expression
1		hmph 10241	. . 3 |- ((J e. Top /\ K e. Top) -> (J ~= K <-> E.f f e. (J Homeo K)))
2		visset 2295	. . . . . 6 |- f e. _V
3		eqid 1884	. . . . . . 7 |- U.J = U.J
4		eqid 1884	. . . . . . 7 |- U.K = U.K
5	3, 4	hmeobc 10239	. . . . . 6 |- ((J e. Top /\ K e. Top /\ f e. _V) -> (f e. (J Homeo K) <-> (f:U.J-1-1-onto->U.K /\ f e. (J Cn K) /\ `'f e. (K Cn J))))
6	2, 5	mp3an3 1180	. . . . 5 |- ((J e. Top /\ K e. Top) -> (f e. (J Homeo K) <-> (f:U.J-1-1-onto->U.K /\ f e. (J Cn K) /\ `'f e. (K Cn J))))
7	3, 4	cncomp 10331	. . . . . . . . . . . . 13 |- (((J e. Comp /\ K e. Top) /\ (f:U.J-onto->U.K /\ f e. (J Cn K))) -> K e. Comp)
8	7	exp43 415	. . . . . . . . . . . 12 |- (J e. Comp -> (K e. Top -> (f:U.J-onto->U.K -> (f e. (J Cn K) -> K e. Comp))))
9	8	com4l 43	. . . . . . . . . . 11 |- (K e. Top -> (f:U.J-onto->U.K -> (f e. (J Cn K) -> (J e. Comp -> K e. Comp))))
10		f1ofo 4643	. . . . . . . . . . 11 |- (f:U.J-1-1-onto->U.K -> f:U.J-onto->U.K)
11	9, 10	syl5 20	. . . . . . . . . 10 |- (K e. Top -> (f:U.J-1-1-onto->U.K -> (f e. (J Cn K) -> (J e. Comp -> K e. Comp))))
12	11	imp32 390	. . . . . . . . 9 |- ((K e. Top /\ (f:U.J-1-1-onto->U.K /\ f e. (J Cn K))) -> (J e. Comp -> K e. Comp))
13	12	adantll 428	. . . . . . . 8 |- (((J e. Top /\ K e. Top) /\ (f:U.J-1-1-onto->U.K /\ f e. (J Cn K))) -> (J e. Comp -> K e. Comp))
14	13	3adantr3 1037	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ (f:U.J-1-1-onto->U.K /\ f e. (J Cn K) /\ `'f e. (K Cn J))) -> (J e. Comp -> K e. Comp))
15	4, 3	cncomp 10331	. . . . . . . . . . . . 13 |- (((K e. Comp /\ J e. Top) /\ (`'f:U.K-onto->U.J /\ `'f e. (K Cn J))) -> J e. Comp)
16	15	exp43 415	. . . . . . . . . . . 12 |- (K e. Comp -> (J e. Top -> (`'f:U.K-onto->U.J -> (`'f e. (K Cn J) -> J e. Comp))))
17	16	com4l 43	. . . . . . . . . . 11 |- (J e. Top -> (`'f:U.K-onto->U.J -> (`'f e. (K Cn J) -> (K e. Comp -> J e. Comp))))
18		f1ocnv 4651	. . . . . . . . . . . 12 |- (f:U.J-1-1-onto->U.K -> `'f:U.K-1-1-onto->U.J)
19		f1ofo 4643	. . . . . . . . . . . 12 |- (`'f:U.K-1-1-onto->U.J -> `'f:U.K-onto->U.J)
20	18, 19	syl 12	. . . . . . . . . . 11 |- (f:U.J-1-1-onto->U.K -> `'f:U.K-onto->U.J)
21	17, 20	syl5 20	. . . . . . . . . 10 |- (J e. Top -> (f:U.J-1-1-onto->U.K -> (`'f e. (K Cn J) -> (K e. Comp -> J e. Comp))))
22	21	imp32 390	. . . . . . . . 9 |- ((J e. Top /\ (f:U.J-1-1-onto->U.K /\ `'f e. (K Cn J))) -> (K e. Comp -> J e. Comp))
23	22	adantlr 429	. . . . . . . 8 |- (((J e. Top /\ K e. Top) /\ (f:U.J-1-1-onto->U.K /\ `'f e. (K Cn J))) -> (K e. Comp -> J e. Comp))
24	23	3adantr2 1036	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ (f:U.J-1-1-onto->U.K /\ f e. (J Cn K) /\ `'f e. (K Cn J))) -> (K e. Comp -> J e. Comp))
25	14, 24	impbid 574	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ (f:U.J-1-1-onto->U.K /\ f e. (J Cn K) /\ `'f e. (K Cn J))) -> (J e. Comp <-> K e. Comp))
26	25	ex 402	. . . . 5 |- ((J e. Top /\ K e. Top) -> ((f:U.J-1-1-onto->U.K /\ f e. (J Cn K) /\ `'f e. (K Cn J)) -> (J e. Comp <-> K e. Comp)))
27	6, 26	sylbid 220	. . . 4 |- ((J e. Top /\ K e. Top) -> (f e. (J Homeo K) -> (J e. Comp <-> K e. Comp)))
28	27	19.23adv 1584	. . 3 |- ((J e. Top /\ K e. Top) -> (E.f f e. (J Homeo K) -> (J e. Comp <-> K e. Comp)))
29	1, 28	sylbid 220	. 2 |- ((J e. Top /\ K e. Top) -> (J ~= K -> (J e. Comp <-> K e. Comp)))
30	29	3impia 1064	1 |- ((J e. Top /\ K e. Top /\ J ~= K) -> (J e. Comp <-> K e. Comp))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   e. wcel 1300  E.wex 1326  _Vcvv 2292  U.cuni 3177   class class class wbr 3338  `'ccnv 3985  -onto->wfo 3996  -1-1-onto->wf1o 3997  (class class class)co 4884  Topctop 8857   Cn ccn 9028   Homeo chomeosm 10230   ~= chomeo 10231  Compccomp 10328

This theorem is referenced by:  reheibor 16025

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-rep 3428  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-3or 859  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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-1o 5177  df-er 5318  df-map 5383  df-en 5427  df-dom 5428  df-fin 5430  df-cn 9030  df-homeo 10232  df-hmph 10233  df-comp 10329

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