Theorem indishmph 21601

Description: Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)

Assertion

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

Proof of Theorem indishmph

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 7964	. 2 |- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
2		f1of 6137	. . . . . . 7 |- ( f : A -1-1-onto-> B -> f : A --> B )
3		f1odm 6141	. . . . . . . . . 10 |- ( f : A -1-1-onto-> B -> dom f = A )
4		vex 3203	. . . . . . . . . . 11 |- f e. _V
5	4	dmex 7099	. . . . . . . . . 10 |- dom f e. _V
6	3, 5	syl6eqelr 2710	. . . . . . . . 9 |- ( f : A -1-1-onto-> B -> A e. _V )
7		f1ofo 6144	. . . . . . . . 9 |- ( f : A -1-1-onto-> B -> f : A -onto-> B )
8		fornex 7135	. . . . . . . . 9 |- ( A e. _V -> ( f : A -onto-> B -> B e. _V ) )
9	6, 7, 8	sylc 65	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> B e. _V )
10	9, 6	elmapd 7871	. . . . . . 7 |- ( f : A -1-1-onto-> B -> ( f e. ( B ^m A ) <-> f : A --> B ) )
11	2, 10	mpbird 247	. . . . . 6 |- ( f : A -1-1-onto-> B -> f e. ( B ^m A ) )
12		indistopon 20805	. . . . . . . 8 |- ( A e. _V -> { (/) , A } e. (TopOn ` A ) )
13	6, 12	syl 17	. . . . . . 7 |- ( f : A -1-1-onto-> B -> { (/) , A } e. (TopOn ` A ) )
14		cnindis 21096	. . . . . . 7 |- ( ( { (/) , A } e. (TopOn ` A ) /\ B e. _V ) -> ( { (/) , A } Cn { (/) , B } ) = ( B ^m A ) )
15	13, 9, 14	syl2anc 693	. . . . . 6 |- ( f : A -1-1-onto-> B -> ( { (/) , A } Cn { (/) , B } ) = ( B ^m A ) )
16	11, 15	eleqtrrd 2704	. . . . 5 |- ( f : A -1-1-onto-> B -> f e. ( { (/) , A } Cn { (/) , B } ) )
17		f1ocnv 6149	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> `' f : B -1-1-onto-> A )
18		f1of 6137	. . . . . . . 8 |- ( `' f : B -1-1-onto-> A -> `' f : B --> A )
19	17, 18	syl 17	. . . . . . 7 |- ( f : A -1-1-onto-> B -> `' f : B --> A )
20	6, 9	elmapd 7871	. . . . . . 7 |- ( f : A -1-1-onto-> B -> ( `' f e. ( A ^m B ) <-> `' f : B --> A ) )
21	19, 20	mpbird 247	. . . . . 6 |- ( f : A -1-1-onto-> B -> `' f e. ( A ^m B ) )
22		indistopon 20805	. . . . . . . 8 |- ( B e. _V -> { (/) , B } e. (TopOn ` B ) )
23	9, 22	syl 17	. . . . . . 7 |- ( f : A -1-1-onto-> B -> { (/) , B } e. (TopOn ` B ) )
24		cnindis 21096	. . . . . . 7 |- ( ( { (/) , B } e. (TopOn ` B ) /\ A e. _V ) -> ( { (/) , B } Cn { (/) , A } ) = ( A ^m B ) )
25	23, 6, 24	syl2anc 693	. . . . . 6 |- ( f : A -1-1-onto-> B -> ( { (/) , B } Cn { (/) , A } ) = ( A ^m B ) )
26	21, 25	eleqtrrd 2704	. . . . 5 |- ( f : A -1-1-onto-> B -> `' f e. ( { (/) , B } Cn { (/) , A } ) )
27		ishmeo 21562	. . . . 5 |- ( f e. ( { (/) , A } Homeo { (/) , B } ) <-> ( f e. ( { (/) , A } Cn { (/) , B } ) /\ `' f e. ( { (/) , B } Cn { (/) , A } ) ) )
28	16, 26, 27	sylanbrc 698	. . . 4 |- ( f : A -1-1-onto-> B -> f e. ( { (/) , A } Homeo { (/) , B } ) )
29		hmphi 21580	. . . 4 |- ( f e. ( { (/) , A } Homeo { (/) , B } ) -> { (/) , A } ~= { (/) , B } )
30	28, 29	syl 17	. . 3 |- ( f : A -1-1-onto-> B -> { (/) , A } ~= { (/) , B } )
31	30	exlimiv 1858	. 2 |- ( E. f f : A -1-1-onto-> B -> { (/) , A } ~= { (/) , B } )
32	1, 31	sylbi 207	1 |- ( A ~~ B -> { (/) , A } ~= { (/) , B } )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {cpr 4179   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   ~~ cen 7952  TopOnctopon 20715   Cn ccn 21028   Homeochmeo 21556   ~= chmph 21557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-map 7859  df-en 7956  df-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558  df-hmph 21559

This theorem is referenced by: (None)