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Theorem 2ndcrest 17196
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 17188 . . 3  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  x  e. 
TopBases )
3 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  A  e.  V )
4 tgrest 16906 . . . . . . . 8  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
52, 3, 4syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
6 restbas 16905 . . . . . . . . 9  |-  ( x  e.  TopBases  ->  ( xt  A )  e.  TopBases )
76ad2antlr 707 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  e.  TopBases )
8 restval 13347 . . . . . . . . . 10  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
92, 3, 8syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
10 1stcrestlem 17194 . . . . . . . . . 10  |-  ( x  ~<_  om  ->  ran  ( y  e.  x  |->  ( y  i^i  A ) )  ~<_  om )
1110adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ran  ( y  e.  x  |->  ( y  i^i  A
) )  ~<_  om )
129, 11eqbrtrd 4059 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  ~<_  om )
13 2ndci 17190 . . . . . . . 8  |-  ( ( ( xt  A )  e.  TopBases  /\  ( xt  A )  ~<_  om )  ->  ( topGen `  ( xt  A
) )  e.  2ndc )
147, 12, 13syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  e.  2ndc )
155, 14eqeltrrd 2371 . . . . . 6  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )t  A
)  e.  2ndc )
16 oveq1 5881 . . . . . . 7  |-  ( (
topGen `  x )  =  J  ->  ( ( topGen `
 x )t  A )  =  ( Jt  A ) )
1716eleq1d 2362 . . . . . 6  |-  ( (
topGen `  x )  =  J  ->  ( (
( topGen `  x )t  A
)  e.  2ndc  <->  ( Jt  A
)  e.  2ndc )
)
1815, 17syl5ibcom 211 . . . . 5  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )  =  J  ->  ( Jt  A )  e.  2ndc )
)
1918expimpd 586 . . . 4  |-  ( ( A  e.  V  /\  x  e.  TopBases )  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  ( Jt  A )  e.  2ndc ) )
2019rexlimdva 2680 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  ( Jt  A
)  e.  2ndc )
)
211, 20syl5bi 208 . 2  |-  ( A  e.  V  ->  ( J  e.  2ndc  ->  ( Jt  A )  e.  2ndc ) )
2221impcom 419 1  |-  ( ( J  e.  2ndc  /\  A  e.  V )  ->  ( Jt  A )  e.  2ndc )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164   class class class wbr 4039    e. cmpt 4093   omcom 4672   ran crn 4706   ` cfv 5271  (class class class)co 5874    ~<_ cdom 6877   ↾t crest 13341   topGenctg 13358   TopBasesctb 16651   2ndcc2ndc 17180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-fin 6883  df-fi 7181  df-card 7588  df-acn 7591  df-rest 13343  df-topgen 13360  df-bases 16654  df-2ndc 17182
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