Theorem hmphdis 21599

Description: Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
hmphdis.1	|- X = U. J

Assertion

Ref	Expression
hmphdis	|- ( J ~= ~P A -> J = ~P X )

Proof of Theorem hmphdis

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwuni 4474	. . . 4 |- J C_ ~P U. J
2		hmphdis.1	. . . . 5 |- X = U. J
3	2	pweqi 4162	. . . 4 |- ~P X = ~P U. J
4	1, 3	sseqtr4i 3638	. . 3 |- J C_ ~P X
5	4	a1i 11	. 2 |- ( J ~= ~P A -> J C_ ~P X )
6		hmph 21579	. . 3 |- ( J ~= ~P A <-> ( J Homeo ~P A ) =/= (/) )
7		n0 3931	. . . 4 |- ( ( J Homeo ~P A ) =/= (/) <-> E. f f e. ( J Homeo ~P A ) )
8		elpwi 4168	. . . . . . 7 |- ( x e. ~P X -> x C_ X )
9		imassrn 5477	. . . . . . . . . . 11 |- ( f " x ) C_ ran f
10		unipw 4918	. . . . . . . . . . . . . . 15 |- U. ~P A = A
11	10	eqcomi 2631	. . . . . . . . . . . . . 14 |- A = U. ~P A
12	2, 11	hmeof1o 21567	. . . . . . . . . . . . 13 |- ( f e. ( J Homeo ~P A ) -> f : X -1-1-onto-> A )
13		f1of 6137	. . . . . . . . . . . . 13 |- ( f : X -1-1-onto-> A -> f : X --> A )
14		frn 6053	. . . . . . . . . . . . 13 |- ( f : X --> A -> ran f C_ A )
15	12, 13, 14	3syl 18	. . . . . . . . . . . 12 |- ( f e. ( J Homeo ~P A ) -> ran f C_ A )
16	15	adantr 481	. . . . . . . . . . 11 |- ( ( f e. ( J Homeo ~P A ) /\ x C_ X ) -> ran f C_ A )
17	9, 16	syl5ss 3614	. . . . . . . . . 10 |- ( ( f e. ( J Homeo ~P A ) /\ x C_ X ) -> ( f " x ) C_ A )
18		vex 3203	. . . . . . . . . . . 12 |- f e. _V
19	18	imaex 7104	. . . . . . . . . . 11 |- ( f " x ) e. _V
20	19	elpw 4164	. . . . . . . . . 10 |- ( ( f " x ) e. ~P A <-> ( f " x ) C_ A )
21	17, 20	sylibr 224	. . . . . . . . 9 |- ( ( f e. ( J Homeo ~P A ) /\ x C_ X ) -> ( f " x ) e. ~P A )
22	2	hmeoopn 21569	. . . . . . . . 9 |- ( ( f e. ( J Homeo ~P A ) /\ x C_ X ) -> ( x e. J <-> ( f " x ) e. ~P A ) )
23	21, 22	mpbird 247	. . . . . . . 8 |- ( ( f e. ( J Homeo ~P A ) /\ x C_ X ) -> x e. J )
24	23	ex 450	. . . . . . 7 |- ( f e. ( J Homeo ~P A ) -> ( x C_ X -> x e. J ) )
25	8, 24	syl5 34	. . . . . 6 |- ( f e. ( J Homeo ~P A ) -> ( x e. ~P X -> x e. J ) )
26	25	ssrdv 3609	. . . . 5 |- ( f e. ( J Homeo ~P A ) -> ~P X C_ J )
27	26	exlimiv 1858	. . . 4 |- ( E. f f e. ( J Homeo ~P A ) -> ~P X C_ J )
28	7, 27	sylbi 207	. . 3 |- ( ( J Homeo ~P A ) =/= (/) -> ~P X C_ J )
29	6, 28	sylbi 207	. 2 |- ( J ~= ~P A -> ~P X C_ J )
30	5, 29	eqssd 3620	1 |- ( J ~= ~P A -> J = ~P X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   ran crn 5115   "cima 5117   -->wf 5884   -1-1-onto->wf1o 5887  (class class class)co 6650   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-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-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-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558  df-hmph 21559

This theorem is referenced by: (None)