Theorem is2ndc 21249

Description: The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
is2ndc	|- ( J e. 2ndc <-> E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) )

Distinct variable group:   x, J

Proof of Theorem is2ndc

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-2ndc 21243	. . 3 |- 2ndc = { j | E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = j ) }
2	1	eleq2i 2693	. 2 |- ( J e. 2ndc <-> J e. { j | E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = j ) } )
3		simpr 477	. . . . 5 |- ( ( x ~<_ om /\ ( topGen ` x ) = J ) -> ( topGen ` x ) = J )
4		fvex 6201	. . . . 5 |- ( topGen ` x ) e. _V
5	3, 4	syl6eqelr 2710	. . . 4 |- ( ( x ~<_ om /\ ( topGen ` x ) = J ) -> J e. _V )
6	5	rexlimivw 3029	. . 3 |- ( E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) -> J e. _V )
7		eqeq2 2633	. . . . 5 |- ( j = J -> ( ( topGen ` x ) = j <-> ( topGen ` x ) = J ) )
8	7	anbi2d 740	. . . 4 |- ( j = J -> ( ( x ~<_ om /\ ( topGen ` x ) = j ) <-> ( x ~<_ om /\ ( topGen ` x ) = J ) ) )
9	8	rexbidv 3052	. . 3 |- ( j = J -> ( E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = j ) <-> E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) ) )
10	6, 9	elab3 3358	. 2 |- ( J e. { j | E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = j ) } <-> E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) )
11	2, 10	bitri 264	1 |- ( J e. 2ndc <-> E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888   omcom 7065   ~<_ cdom 7953   topGenctg 16098   TopBasesctb 20749   2ndcc2ndc 21241

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-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896  df-2ndc 21243

This theorem is referenced by:  2ndctop  21250  2ndci  21251  2ndcsb  21252  2ndcredom  21253  2ndc1stc  21254  2ndcrest  21257  2ndcctbss  21258  2ndcdisj  21259  2ndcomap  21261  2ndcsep  21262  dis2ndc  21263  tx2ndc  21454