Theorem eqindhome 11047

Description: Equinumerous sets equipped with their indiscrete topologies are homeomorph (which means in particular that a segment is homeomorph to a circle contrary to what Wikipedia claims).

Hypotheses

Ref	Expression
eqindhome.1	|- A e. C
eqindhome.2	|- B e. D

Assertion

Ref	Expression
eqindhome	|- (A ~~ B -> {(/), A} ~= {(/), B})

Proof of Theorem eqindhome

Step	Hyp	Ref	Expression
1		id 59	. . . . 5 |- (f:A-1-1-onto->B -> f:A-1-1-onto->B)
2		imaeq2 3492	. . . . . . . . . 10 |- (x = (/) -> (f"x) = (f"(/)))
3	2	eqeq1d 1526	. . . . . . . . 9 |- (x = (/) -> ((f"x) = (/) <-> (f"(/)) = (/)))
4		ima0 3512	. . . . . . . . . 10 |- (f"(/)) = (/)
5	4	a1i 8	. . . . . . . . 9 |- (f:A-1-1-onto->B -> (f"(/)) = (/))
6	3, 5	syl5cbir 209	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = (/) -> (f"x) = (/)))
7		imaeq2 3492	. . . . . . . . . 10 |- (x = A -> (f"x) = (f"A))
8	7	eqeq1d 1526	. . . . . . . . 9 |- (x = A -> ((f"x) = B <-> (f"A) = B))
9		f1ofo 3803	. . . . . . . . . 10 |- (f:A-1-1-onto->B -> f:A-onto->B)
10		foima 3784	. . . . . . . . . 10 |- (f:A-onto->B -> (f"A) = B)
11	9, 10	syl 10	. . . . . . . . 9 |- (f:A-1-1-onto->B -> (f"A) = B)
12	8, 11	syl5cbir 209	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = A -> (f"x) = B))
13	6, 12	orim12d 568	. . . . . . 7 |- (f:A-1-1-onto->B -> ((x = (/) \/ x = A) -> ((f"x) = (/) \/ (f"x) = B)))
14		visset 1859	. . . . . . . 8 |- x e. V
15	14	elpr 2482	. . . . . . 7 |- (x e. {(/), A} <-> (x = (/) \/ x = A))
16		visset 1859	. . . . . . . . 9 |- f e. V
17		imaexg 3508	. . . . . . . . 9 |- (f e. V -> (f"x) e. V)
18	16, 17	ax-mp 7	. . . . . . . 8 |- (f"x) e. V
19	18	elpr 2482	. . . . . . 7 |- ((f"x) e. {(/), B} <-> ((f"x) = (/) \/ (f"x) = B))
20	13, 15, 19	3imtr4g 556	. . . . . 6 |- (f:A-1-1-onto->B -> (x e. {(/), A} -> (f"x) e. {(/), B}))
21	20	r19.21aiv 1759	. . . . 5 |- (f:A-1-1-onto->B -> A.x e. {(/), A} (f"x) e. {(/), B})
22		imaeq2 3492	. . . . . . . . . 10 |- (x = (/) -> (`'f"x) = (`'f"(/)))
23		ima0 3512	. . . . . . . . . 10 |- (`'f"(/)) = (/)
24	22, 23	syl6eq 1566	. . . . . . . . 9 |- (x = (/) -> (`'f"x) = (/))
25	24	a1i 8	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = (/) -> (`'f"x) = (/)))
26		imaeq2 3492	. . . . . . . . . 10 |- (x = B -> (`'f"x) = (`'f"B))
27	26	eqeq1d 1526	. . . . . . . . 9 |- (x = B -> ((`'f"x) = A <-> (`'f"B) = A))
28		f1of 3797	. . . . . . . . . 10 |- (f:A-1-1-onto->B -> f:A-->B)
29		fimacnv 3924	. . . . . . . . . 10 |- (f:A-->B -> (`'f"B) = A)
30	28, 29	syl 10	. . . . . . . . 9 |- (f:A-1-1-onto->B -> (`'f"B) = A)
31	27, 30	syl5cbir 209	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = B -> (`'f"x) = A))
32	25, 31	orim12d 568	. . . . . . 7 |- (f:A-1-1-onto->B -> ((x = (/) \/ x = B) -> ((`'f"x) = (/) \/ (`'f"x) = A)))
33	14	elpr 2482	. . . . . . 7 |- (x e. {(/), B} <-> (x = (/) \/ x = B))
34	16	cnvex 3625	. . . . . . . . 9 |- `'f e. V
35		imaexg 3508	. . . . . . . . 9 |- (`'f e. V -> (`'f"x) e. V)
36	34, 35	ax-mp 7	. . . . . . . 8 |- (`'f"x) e. V
37	36	elpr 2482	. . . . . . 7 |- ((`'f"x) e. {(/), A} <-> ((`'f"x) = (/) \/ (`'f"x) = A))
38	32, 33, 37	3imtr4g 556	. . . . . 6 |- (f:A-1-1-onto->B -> (x e. {(/), B} -> (`'f"x) e. {(/), A}))
39	38	r19.21aiv 1759	. . . . 5 |- (f:A-1-1-onto->B -> A.x e. {(/), B} (`'f"x) e. {(/), A})
40	1, 21, 39	3jca 825	. . . 4 |- (f:A-1-1-onto->B -> (f:A-1-1-onto->B /\ A.x e. {(/), A} (f"x) e. {(/), B} /\ A.x e. {(/), B} (`'f"x) e. {(/), A}))
41		indistop 7860	. . . . 5 |- {(/), A} e. Top
42		indistop 7860	. . . . 5 |- {(/), B} e. Top
43		0ex 2785	. . . . . . . 8 |- (/) e. V
44		eqindhome.1	. . . . . . . . 9 |- A e. C
45	44	elisseti 1864	. . . . . . . 8 |- A e. V
46	43, 45	unipr 2581	. . . . . . 7 |- U.{(/), A} = ((/) u. A)
47		uncom 2228	. . . . . . 7 |- ((/) u. A) = (A u. (/))
48		un0 2350	. . . . . . 7 |- (A u. (/)) = A
49	46, 47, 48	3eqtrri 1543	. . . . . 6 |- A = U.{(/), A}
50		eqindhome.2	. . . . . . . . 9 |- B e. D
51	50	elisseti 1864	. . . . . . . 8 |- B e. V
52	43, 51	unipr 2581	. . . . . . 7 |- U.{(/), B} = ((/) u. B)
53		uncom 2228	. . . . . . 7 |- ((/) u. B) = (B u. (/))
54		un0 2350	. . . . . . 7 |- (B u. (/)) = B
55	52, 53, 54	3eqtrri 1543	. . . . . 6 |- B = U.{(/), B}
56	49, 55	ishomeo 11023	. . . . 5 |- (({(/), A} e. Top /\ {(/), B} e. Top /\ f e. V) -> (f e. ({(/), A} Homeo {(/), B}) <-> (f:A-1-1-onto->B /\ A.x e. {(/), A} (f"x) e. {(/), B} /\ A.x e. {(/), B} (`'f"x) e. {(/), A})))
57	41, 42, 16, 56	mp3an 922	. . . 4 |- (f e. ({(/), A} Homeo {(/), B}) <-> (f:A-1-1-onto->B /\ A.x e. {(/), A} (f"x) e. {(/), B} /\ A.x e. {(/), B} (`'f"x) e. {(/), A}))
58	40, 57	sylibr 198	. . 3 |- (f:A-1-1-onto->B -> f e. ({(/), A} Homeo {(/), B}))
59	58	19.22i 1076	. 2 |- (E.f f:A-1-1-onto->B -> E.f f e. ({(/), A} Homeo {(/), B}))
60	51	bren 4518	. 2 |- (A ~~ B <-> E.f f:A-1-1-onto->B)
61		hmph 11030	. . 3 |- (({(/), A} e. Top /\ {(/), B} e. Top) -> ({(/), A} ~= {(/), B} <-> E.f f e. ({(/), A} Homeo {(/), B})))
62	41, 42, 61	mp2an 701	. 2 |- ({(/), A} ~= {(/), B} <-> E.f f e. ({(/), A} Homeo {(/), B}))
63	59, 60, 62	3imtr4i 217	1 |- (A ~~ B -> {(/), A} ~= {(/), B})

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   /\ w3a 781   = wceq 992   e. wcel 994  E.wex 1016  A.wral 1691  Vcvv 1857   u. cun 2097  (/)c0 2332  {cpr 2468  U.cuni 2569   class class class wbr 2692  `'ccnv 3250  "cima 3254  -->wf 3259  -onto->wfo 3261  -1-1-onto->wf1o 3262  (class class class)co 4021   ~~ cen 4505  Topctop 7800   Homeo chomeosm 11019   ~= chomeo 11020

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-en 4509  df-top 7804  df-homeo 11021  df-hmph 11029

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