Theorem isconn 21216

Description: The predicate J is a connected topology . (Contributed by FL, 17-Nov-2008.)

Hypothesis

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

Assertion

Ref	Expression
isconn	|- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) = { (/) , X } ) )

Proof of Theorem isconn

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( j = J -> j = J )
2		fveq2 6191	. . . 4 |- ( j = J -> ( Clsd ` j ) = ( Clsd ` J ) )
3	1, 2	ineq12d 3815	. . 3 |- ( j = J -> ( j i^i ( Clsd ` j ) ) = ( J i^i ( Clsd ` J ) ) )
4		unieq 4444	. . . . 5 |- ( j = J -> U. j = U. J )
5		isconn.1	. . . . 5 |- X = U. J
6	4, 5	syl6eqr 2674	. . . 4 |- ( j = J -> U. j = X )
7	6	preq2d 4275	. . 3 |- ( j = J -> { (/) , U. j } = { (/) , X } )
8	3, 7	eqeq12d 2637	. 2 |- ( j = J -> ( ( j i^i ( Clsd ` j ) ) = { (/) , U. j } <-> ( J i^i ( Clsd ` J ) ) = { (/) , X } ) )
9		df-conn 21215	. 2 |- Conn = { j e. Top | ( j i^i ( Clsd ` j ) ) = { (/) , U. j } }
10	8, 9	elrab2 3366	1 |- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) = { (/) , X } ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   (/)c0 3915   {cpr 4179   U.cuni 4436   ` cfv 5888   Topctop 20698   Clsdccld 20820  Conncconn 21214

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-conn 21215

This theorem is referenced by:  isconn2  21217  connclo  21218  conndisj  21219  conntop  21220