MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ndcrest Structured version   Unicode version

Theorem 2ndcrest 19058
Description: A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndcrest  |-  ( ( J  e.  2ndc  /\  A  e.  V )  ->  ( Jt  A )  e.  2ndc )

Proof of Theorem 2ndcrest
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2ndc 19050 . . 3  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  x  e. 
TopBases )
3 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  A  e.  V )
4 tgrest 18763 . . . . . . . 8  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
52, 3, 4syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
6 restbas 18762 . . . . . . . . 9  |-  ( x  e.  TopBases  ->  ( xt  A )  e.  TopBases )
76ad2antlr 726 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  e.  TopBases )
8 restval 14365 . . . . . . . . . 10  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
92, 3, 8syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
10 1stcrestlem 19056 . . . . . . . . . 10  |-  ( x  ~<_  om  ->  ran  ( y  e.  x  |->  ( y  i^i  A ) )  ~<_  om )
1110adantl 466 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ran  ( y  e.  x  |->  ( y  i^i  A
) )  ~<_  om )
129, 11eqbrtrd 4312 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  ~<_  om )
13 2ndci 19052 . . . . . . . 8  |-  ( ( ( xt  A )  e.  TopBases  /\  ( xt  A )  ~<_  om )  ->  ( topGen `  ( xt  A
) )  e.  2ndc )
147, 12, 13syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  e.  2ndc )
155, 14eqeltrrd 2518 . . . . . 6  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )t  A
)  e.  2ndc )
16 oveq1 6098 . . . . . . 7  |-  ( (
topGen `  x )  =  J  ->  ( ( topGen `
 x )t  A )  =  ( Jt  A ) )
1716eleq1d 2509 . . . . . 6  |-  ( (
topGen `  x )  =  J  ->  ( (
( topGen `  x )t  A
)  e.  2ndc  <->  ( Jt  A
)  e.  2ndc )
)
1815, 17syl5ibcom 220 . . . . 5  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )  =  J  ->  ( Jt  A )  e.  2ndc )
)
1918expimpd 603 . . . 4  |-  ( ( A  e.  V  /\  x  e.  TopBases )  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  ( Jt  A )  e.  2ndc ) )
2019rexlimdva 2841 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  ( Jt  A
)  e.  2ndc )
)
211, 20syl5bi 217 . 2  |-  ( A  e.  V  ->  ( J  e.  2ndc  ->  ( Jt  A )  e.  2ndc ) )
2221impcom 430 1  |-  ( ( J  e.  2ndc  /\  A  e.  V )  ->  ( Jt  A )  e.  2ndc )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716    i^i cin 3327   class class class wbr 4292    e. cmpt 4350   ran crn 4841   ` cfv 5418  (class class class)co 6091   omcom 6476    ~<_ cdom 7308   ↾t crest 14359   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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-fin 7314  df-fi 7661  df-card 8109  df-acn 8112  df-rest 14361  df-topgen 14382  df-bases 18505  df-2ndc 19044
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator