Theorem ist1 21125

Description: The predicate J is T₁. (Contributed by FL, 18-Jun-2007.)

Hypothesis

Ref	Expression
ist0.1	|- X = U. J

Assertion

Ref	Expression
ist1	|- ( J e. Fre <-> ( J e. Top /\ A. a e. X { a } e. ( Clsd ` J ) ) )

Distinct variable group:   J, a

Allowed substitution hint:   X( a)

Proof of Theorem ist1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4444	. . . . . 6 |- ( x = J -> U. x = U. J )
2		ist0.1	. . . . . 6 |- X = U. J
3	1, 2	syl6eqr 2674	. . . . 5 |- ( x = J -> U. x = X )
4	3	eleq2d 2687	. . . 4 |- ( x = J -> ( a e. U. x <-> a e. X ) )
5		fveq2 6191	. . . . 5 |- ( x = J -> ( Clsd ` x ) = ( Clsd ` J ) )
6	5	eleq2d 2687	. . . 4 |- ( x = J -> ( { a } e. ( Clsd ` x ) <-> { a } e. ( Clsd ` J ) ) )
7	4, 6	imbi12d 334	. . 3 |- ( x = J -> ( ( a e. U. x -> { a } e. ( Clsd ` x ) ) <-> ( a e. X -> { a } e. ( Clsd ` J ) ) ) )
8	7	ralbidv2 2984	. 2 |- ( x = J -> ( A. a e. U. x { a } e. ( Clsd ` x ) <-> A. a e. X { a } e. ( Clsd ` J ) ) )
9		df-t1 21118	. 2 |- Fre = { x e. Top | A. a e. U. x { a } e. ( Clsd ` x ) }
10	8, 9	elrab2 3366	1 |- ( J e. Fre <-> ( J e. Top /\ A. a e. X { a } e. ( Clsd ` J ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   U.cuni 4436   ` cfv 5888   Topctop 20698   Clsdccld 20820   Frect1 21111

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-ral 2917  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-t1 21118

This theorem is referenced by:  t1sncld  21130  t1ficld  21131  t1top  21134  ist1-2  21151  cnt1  21154  ordtt1  21183  qtopt1  29902  onint1  32448