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Theorem 2ndci 20075
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 2458 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  =  (
topGen `  B ) )
4 breq1 4459 . . . . 5  |-  ( x  =  B  ->  (
x  ~<_  om  <->  B  ~<_  om )
)
5 fveq2 5872 . . . . . 6  |-  ( x  =  B  ->  ( topGen `
 x )  =  ( topGen `  B )
)
65eqeq1d 2459 . . . . 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 3210 . . 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 20073 . 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 1395    e. wcel 1819   E.wrex 2808   class class class wbr 4456   ` cfv 5594   omcom 6699    ~<_ cdom 7533   topGenctg 14855   TopBasesctb 19525   2ndcc2ndc 20065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-2ndc 20067
This theorem is referenced by:  2ndcrest  20081  2ndcomap  20085  dis2ndc  20087  dis1stc  20126  tx2ndc  20278  met2ndci  21151  re2ndc  21432
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