Theorem opncldf2 20889

Description: The values 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
opncldf2	|- ( ( J e. Top /\ A e. J ) -> ( F ` A ) = ( X \ A ) )

Distinct variable groups:   u, A   u, J   u, X

Allowed substitution hint:   F( u)

Proof of Theorem opncldf2

Step	Hyp	Ref	Expression
1		simpr 477	. 2 |- ( ( J e. Top /\ A e. J ) -> A e. J )
2		opncldf.1	. . 3 |- X = U. J
3	2	opncld 20837	. 2 |- ( ( J e. Top /\ A e. J ) -> ( X \ A ) e. ( Clsd ` J ) )
4		difeq2 3722	. . 3 |- ( u = A -> ( X \ u ) = ( X \ A ) )
5		opncldf.2	. . 3 |- F = ( u e. J |-> ( X \ u ) )
6	4, 5	fvmptg 6280	. 2 |- ( ( A e. J /\ ( X \ A ) e. ( Clsd ` J ) ) -> ( F ` A ) = ( X \ A ) )
7	1, 3, 6	syl2anc 693	1 |- ( ( J e. Top /\ A e. J ) -> ( F ` A ) = ( X \ A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   U.cuni 4436   |-> cmpt 4729   ` 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-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

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-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-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-cld 20823

This theorem is referenced by: (None)