Theorem eqindhome 14895

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 73	. . . . 5 |- (f:A-1-1-onto->B -> f:A-1-1-onto->B)
2		imaeq2 4260	. . . . . . . . . 10 |- (x = (/) -> (f"x) = (f"(/)))
3	2	eqeq1d 1892	. . . . . . . . 9 |- (x = (/) -> ((f"x) = (/) <-> (f"(/)) = (/)))
4		ima0 4283	. . . . . . . . . 10 |- (f"(/)) = (/)
5	4	a1i 8	. . . . . . . . 9 |- (f:A-1-1-onto->B -> (f"(/)) = (/))
6	3, 5	syl5cbir 228	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = (/) -> (f"x) = (/)))
7		imaeq2 4260	. . . . . . . . . 10 |- (x = A -> (f"x) = (f"A))
8	7	eqeq1d 1892	. . . . . . . . 9 |- (x = A -> ((f"x) = B <-> (f"A) = B))
9		f1ofo 4643	. . . . . . . . . 10 |- (f:A-1-1-onto->B -> f:A-onto->B)
10		foima 4622	. . . . . . . . . 10 |- (f:A-onto->B -> (f"A) = B)
11	9, 10	syl 12	. . . . . . . . 9 |- (f:A-1-1-onto->B -> (f"A) = B)
12	8, 11	syl5cbir 228	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = A -> (f"x) = B))
13	6, 12	orim12d 624	. . . . . . 7 |- (f:A-1-1-onto->B -> ((x = (/) \/ x = A) -> ((f"x) = (/) \/ (f"x) = B)))
14		visset 2295	. . . . . . . 8 |- x e. _V
15	14	elpr 3061	. . . . . . 7 |- (x e. {(/), A} <-> (x = (/) \/ x = A))
16		visset 2295	. . . . . . . . 9 |- f e. _V
17		imaexg 4279	. . . . . . . . 9 |- (f e. _V -> (f"x) e. _V)
18	16, 17	ax-mp 7	. . . . . . . 8 |- (f"x) e. _V
19	18	elpr 3061	. . . . . . 7 |- ((f"x) e. {(/), B} <-> ((f"x) = (/) \/ (f"x) = B))
20	13, 15, 19	3imtr4g 612	. . . . . 6 |- (f:A-1-1-onto->B -> (x e. {(/), A} -> (f"x) e. {(/), B}))
21	20	r19.21aiv 2175	. . . . 5 |- (f:A-1-1-onto->B -> A.x e. {(/), A} (f"x) e. {(/), B})
22		imaeq2 4260	. . . . . . . . . 10 |- (x = (/) -> (`'f"x) = (`'f"(/)))
23		ima0 4283	. . . . . . . . . 10 |- (`'f"(/)) = (/)
24	22, 23	syl6eq 1944	. . . . . . . . 9 |- (x = (/) -> (`'f"x) = (/))
25	24	a1i 8	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = (/) -> (`'f"x) = (/)))
26		imaeq2 4260	. . . . . . . . . 10 |- (x = B -> (`'f"x) = (`'f"B))
27	26	eqeq1d 1892	. . . . . . . . 9 |- (x = B -> ((`'f"x) = A <-> (`'f"B) = A))
28		f1of 4635	. . . . . . . . . 10 |- (f:A-1-1-onto->B -> f:A-->B)
29		fimacnv 4783	. . . . . . . . . 10 |- (f:A-->B -> (`'f"B) = A)
30	28, 29	syl 12	. . . . . . . . 9 |- (f:A-1-1-onto->B -> (`'f"B) = A)
31	27, 30	syl5cbir 228	. . . . . . . 8 |- (f:A-1-1-onto->B -> (x = B -> (`'f"x) = A))
32	25, 31	orim12d 624	. . . . . . 7 |- (f:A-1-1-onto->B -> ((x = (/) \/ x = B) -> ((`'f"x) = (/) \/ (`'f"x) = A)))
33	14	elpr 3061	. . . . . . 7 |- (x e. {(/), B} <-> (x = (/) \/ x = B))
34	16	cnvex 4425	. . . . . . . . 9 |- `'f e. _V
35		imaexg 4279	. . . . . . . . 9 |- (`'f e. _V -> (`'f"x) e. _V)
36	34, 35	ax-mp 7	. . . . . . . 8 |- (`'f"x) e. _V
37	36	elpr 3061	. . . . . . 7 |- ((`'f"x) e. {(/), A} <-> ((`'f"x) = (/) \/ (`'f"x) = A))
38	32, 33, 37	3imtr4g 612	. . . . . 6 |- (f:A-1-1-onto->B -> (x e. {(/), B} -> (`'f"x) e. {(/), A}))
39	38	r19.21aiv 2175	. . . . 5 |- (f:A-1-1-onto->B -> A.x e. {(/), B} (`'f"x) e. {(/), A})
40	1, 21, 39	3jca 1050	. . . 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 8918	. . . . 5 |- {(/), A} e. Top
42		indistop 8918	. . . . 5 |- {(/), B} e. Top
43		0ex 3446	. . . . . . . 8 |- (/) e. _V
44		eqindhome.1	. . . . . . . . 9 |- A e. C
45	44	elisseti 2301	. . . . . . . 8 |- A e. _V
46	43, 45	unipr 3191	. . . . . . 7 |- U.{(/), A} = ((/) u. A)
47		uncom 2744	. . . . . . 7 |- ((/) u. A) = (A u. (/))
48		un0 2896	. . . . . . 7 |- (A u. (/)) = A
49	46, 47, 48	3eqtrri 1913	. . . . . 6 |- A = U.{(/), A}
50		eqindhome.2	. . . . . . . . 9 |- B e. D
51	50	elisseti 2301	. . . . . . . 8 |- B e. _V
52	43, 51	unipr 3191	. . . . . . 7 |- U.{(/), B} = ((/) u. B)
53		uncom 2744	. . . . . . 7 |- ((/) u. B) = (B u. (/))
54		un0 2896	. . . . . . 7 |- (B u. (/)) = B
55	52, 53, 54	3eqtrri 1913	. . . . . 6 |- B = U.{(/), B}
56	49, 55	ishomeo 10235	. . . . 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 1191	. . . 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 217	. . 3 |- (f:A-1-1-onto->B -> f e. ({(/), A} Homeo {(/), B}))
59	58	eximi 1387	. 2 |- (E.f f:A-1-1-onto->B -> E.f f e. ({(/), A} Homeo {(/), B}))
60	51	bren 5436	. 2 |- (A ~~ B <-> E.f f:A-1-1-onto->B)
61		hmph 10241	. . 3 |- (({(/), A} e. Top /\ {(/), B} e. Top) -> ({(/), A} ~= {(/), B} <-> E.f f e. ({(/), A} Homeo {(/), B})))
62	41, 42, 61	mp2an 761	. 2 |- ({(/), A} ~= {(/), B} <-> E.f f e. ({(/), A} Homeo {(/), B}))
63	59, 60, 62	3imtr4i 236	1 |- (A ~~ B -> {(/), A} ~= {(/), B})

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  _Vcvv 2292   u. cun 2591  (/)c0 2875  {cpr 3045  U.cuni 3177   class class class wbr 3338  `'ccnv 3985  "cima 3989  -->wf 3994  -onto->wfo 3996  -1-1-onto->wf1o 3997  (class class class)co 4884   ~~ cen 5423  Topctop 8857   Homeo chomeosm 10230   ~= chomeo 10231

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-en 5427  df-top 8861  df-homeo 10232  df-hmph 10233

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