Theorem opncldf3 20890

Description: The values of the converse/inverse of the open-closed bijection. (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
opncldf3	|- ( B e. ( Clsd ` J ) -> ( `' F ` B ) = ( X \ B ) )

Distinct variable groups:   u, J   u, X

Allowed substitution hints:   B( u)   F( u)

Proof of Theorem opncldf3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldrcl 20830	. . . 4 |- ( B e. ( Clsd ` J ) -> J e. Top )
2		opncldf.1	. . . . . 6 |- X = U. J
3		opncldf.2	. . . . . 6 |- F = ( u e. J |-> ( X \ u ) )
4	2, 3	opncldf1 20888	. . . . 5 |- ( J e. Top -> ( F : J -1-1-onto-> ( Clsd ` J ) /\ `' F = ( x e. ( Clsd ` J ) |-> ( X \ x ) ) ) )
5	4	simprd 479	. . . 4 |- ( J e. Top -> `' F = ( x e. ( Clsd ` J ) |-> ( X \ x ) ) )
6	1, 5	syl 17	. . 3 |- ( B e. ( Clsd ` J ) -> `' F = ( x e. ( Clsd ` J ) |-> ( X \ x ) ) )
7	6	fveq1d 6193	. 2 |- ( B e. ( Clsd ` J ) -> ( `' F ` B ) = ( ( x e. ( Clsd ` J ) |-> ( X \ x ) ) ` B ) )
8	2	cldopn 20835	. . 3 |- ( B e. ( Clsd ` J ) -> ( X \ B ) e. J )
9		difeq2 3722	. . . 4 |- ( x = B -> ( X \ x ) = ( X \ B ) )
10		eqid 2622	. . . 4 |- ( x e. ( Clsd ` J ) |-> ( X \ x ) ) = ( x e. ( Clsd ` J ) |-> ( X \ x ) )
11	9, 10	fvmptg 6280	. . 3 |- ( ( B e. ( Clsd ` J ) /\ ( X \ B ) e. J ) -> ( ( x e. ( Clsd ` J ) |-> ( X \ x ) ) ` B ) = ( X \ B ) )
12	8, 11	mpdan 702	. 2 |- ( B e. ( Clsd ` J ) -> ( ( x e. ( Clsd ` J ) |-> ( X \ x ) ) ` B ) = ( X \ B ) )
13	7, 12	eqtrd 2656	1 |- ( B e. ( Clsd ` J ) -> ( `' F ` B ) = ( X \ B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   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: (None)