Theorem opncldf1 20888

Description: A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
opncldf.1	|- X = U. J
opncldf.2	|- F = ( u e. J |-> ( X \ u ) )

Assertion

Ref	Expression
opncldf1	|- ( J e. Top -> ( F : J -1-1-onto-> ( Clsd ` J ) /\ `' F = ( x e. ( Clsd ` J ) |-> ( X \ x ) ) ) )

Distinct variable groups:   x, F   x, u, J   u, X, x

Allowed substitution hint:   F( u)

Proof of Theorem opncldf1

Step	Hyp	Ref	Expression
1		opncldf.2	. 2 |- F = ( u e. J |-> ( X \ u ) )
2		opncldf.1	. . 3 |- X = U. J
3	2	opncld 20837	. 2 |- ( ( J e. Top /\ u e. J ) -> ( X \ u ) e. ( Clsd ` J ) )
4	2	cldopn 20835	. . 3 |- ( x e. ( Clsd ` J ) -> ( X \ x ) e. J )
5	4	adantl 482	. 2 |- ( ( J e. Top /\ x e. ( Clsd ` J ) ) -> ( X \ x ) e. J )
6	2	cldss 20833	. . . . . . 7 |- ( x e. ( Clsd ` J ) -> x C_ X )
7	6	ad2antll 765	. . . . . 6 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> x C_ X )
8		dfss4 3858	. . . . . 6 |- ( x C_ X <-> ( X \ ( X \ x ) ) = x )
9	7, 8	sylib 208	. . . . 5 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> ( X \ ( X \ x ) ) = x )
10	9	eqcomd 2628	. . . 4 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> x = ( X \ ( X \ x ) ) )
11		difeq2 3722	. . . . 5 |- ( u = ( X \ x ) -> ( X \ u ) = ( X \ ( X \ x ) ) )
12	11	eqeq2d 2632	. . . 4 |- ( u = ( X \ x ) -> ( x = ( X \ u ) <-> x = ( X \ ( X \ x ) ) ) )
13	10, 12	syl5ibrcom 237	. . 3 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> ( u = ( X \ x ) -> x = ( X \ u ) ) )
14	2	eltopss 20712	. . . . . . 7 |- ( ( J e. Top /\ u e. J ) -> u C_ X )
15	14	adantrr 753	. . . . . 6 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> u C_ X )
16		dfss4 3858	. . . . . 6 |- ( u C_ X <-> ( X \ ( X \ u ) ) = u )
17	15, 16	sylib 208	. . . . 5 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> ( X \ ( X \ u ) ) = u )
18	17	eqcomd 2628	. . . 4 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> u = ( X \ ( X \ u ) ) )
19		difeq2 3722	. . . . 5 |- ( x = ( X \ u ) -> ( X \ x ) = ( X \ ( X \ u ) ) )
20	19	eqeq2d 2632	. . . 4 |- ( x = ( X \ u ) -> ( u = ( X \ x ) <-> u = ( X \ ( X \ u ) ) ) )
21	18, 20	syl5ibrcom 237	. . 3 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> ( x = ( X \ u ) -> u = ( X \ x ) ) )
22	13, 21	impbid 202	. 2 |- ( ( J e. Top /\ ( u e. J /\ x e. ( Clsd ` J ) ) ) -> ( u = ( X \ x ) <-> x = ( X \ u ) ) )
23	1, 3, 5, 22	f1ocnv2d 6886	1 |- ( J e. Top -> ( F : J -1-1-onto-> ( Clsd ` J ) /\ `' F = ( x e. ( Clsd ` J ) |-> ( X \ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888   Topctop 20698   Clsdccld 20820

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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823

This theorem is referenced by:  opncldf3  20890  cmpfi  21211