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Theorem 2ndci 19817
Description: A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndci  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  e.  2ndc )

Proof of Theorem 2ndci
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  B  e.  TopBases )
2 simpr 461 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  B  ~<_  om )
3 eqidd 2468 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  =  (
topGen `  B ) )
4 breq1 4456 . . . . 5  |-  ( x  =  B  ->  (
x  ~<_  om  <->  B  ~<_  om )
)
5 fveq2 5872 . . . . . 6  |-  ( x  =  B  ->  ( topGen `
 x )  =  ( topGen `  B )
)
65eqeq1d 2469 . . . . 5  |-  ( x  =  B  ->  (
( topGen `  x )  =  ( topGen `  B
)  <->  ( topGen `  B
)  =  ( topGen `  B ) ) )
74, 6anbi12d 710 . . . 4  |-  ( x  =  B  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) )  <->  ( B  ~<_  om  /\  ( topGen `  B
)  =  ( topGen `  B ) ) ) )
87rspcev 3219 . . 3  |-  ( ( B  e.  TopBases  /\  ( B  ~<_  om  /\  ( topGen `
 B )  =  ( topGen `  B )
) )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
91, 2, 3, 8syl12anc 1226 . 2  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
10 is2ndc 19815 . 2  |-  ( (
topGen `  B )  e. 
2ndc 
<->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  ( topGen `  B )
) )
119, 10sylibr 212 1  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  e.  2ndc )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   ` cfv 5594   omcom 6695    ~<_ cdom 7526   topGenctg 14710   TopBasesctb 19267   2ndcc2ndc 19807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-2ndc 19809
This theorem is referenced by:  2ndcrest  19823  2ndcomap  19827  dis2ndc  19829  dis1stc  19868  tx2ndc  20020  met2ndci  20893  re2ndc  21174
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