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Theorem 2ndci 17174
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 443 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  B  e.  TopBases )
2 simpr 447 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  B  ~<_  om )
3 eqidd 2284 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  =  (
topGen `  B ) )
4 breq1 4026 . . . . 5  |-  ( x  =  B  ->  (
x  ~<_  om  <->  B  ~<_  om )
)
5 fveq2 5525 . . . . . 6  |-  ( x  =  B  ->  ( topGen `
 x )  =  ( topGen `  B )
)
65eqeq1d 2291 . . . . 5  |-  ( x  =  B  ->  (
( topGen `  x )  =  ( topGen `  B
)  <->  ( topGen `  B
)  =  ( topGen `  B ) ) )
74, 6anbi12d 691 . . . 4  |-  ( x  =  B  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) )  <->  ( B  ~<_  om  /\  ( topGen `  B
)  =  ( topGen `  B ) ) ) )
87rspcev 2884 . . 3  |-  ( ( B  e.  TopBases  /\  ( B  ~<_  om  /\  ( topGen `
 B )  =  ( topGen `  B )
) )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
91, 2, 3, 8syl12anc 1180 . 2  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
10 is2ndc 17172 . 2  |-  ( (
topGen `  B )  e. 
2ndc 
<->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  ( topGen `  B )
) )
119, 10sylibr 203 1  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  e.  2ndc )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   omcom 4656   ` cfv 5255    ~<_ cdom 6861   topGenctg 13342   TopBasesctb 16635   2ndcc2ndc 17164
This theorem is referenced by:  2ndcrest  17180  2ndcomap  17184  dis2ndc  17186  dis1stc  17225  tx2ndc  17345  met2ndci  18068  re2ndc  18307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-2ndc 17166
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