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Theorem 2ndci 19052
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 2444 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  =  (
topGen `  B ) )
4 breq1 4295 . . . . 5  |-  ( x  =  B  ->  (
x  ~<_  om  <->  B  ~<_  om )
)
5 fveq2 5691 . . . . . 6  |-  ( x  =  B  ->  ( topGen `
 x )  =  ( topGen `  B )
)
65eqeq1d 2451 . . . . 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 3073 . . 3  |-  ( ( B  e.  TopBases  /\  ( B  ~<_  om  /\  ( topGen `
 B )  =  ( topGen `  B )
) )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
91, 2, 3, 8syl12anc 1216 . 2  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
10 is2ndc 19050 . 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 1369    e. wcel 1756   E.wrex 2716   class class class wbr 4292   ` cfv 5418   omcom 6476    ~<_ cdom 7308   topGenctg 14376   TopBasesctb 18502   2ndcc2ndc 19042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-2ndc 19044
This theorem is referenced by:  2ndcrest  19058  2ndcomap  19062  dis2ndc  19064  dis1stc  19103  tx2ndc  19224  met2ndci  20097  re2ndc  20378
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