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Theorem 2ndcrest 19823
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 19815 . . 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 19528 . . . . . . . 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 19527 . . . . . . . . 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 14699 . . . . . . . . . 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 19821 . . . . . . . . . 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 4473 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  ~<_  om )
13 2ndci 19817 . . . . . . . 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 2556 . . . . . 6  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )t  A
)  e.  2ndc )
16 oveq1 6302 . . . . . . 7  |-  ( (
topGen `  x )  =  J  ->  ( ( topGen `
 x )t  A )  =  ( Jt  A ) )
1716eleq1d 2536 . . . . . 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 2959 . . 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 1379    e. wcel 1767   E.wrex 2818    i^i cin 3480   class class class wbr 4453    |-> cmpt 4511   ran crn 5006   ` cfv 5594  (class class class)co 6295   omcom 6695    ~<_ cdom 7526   ↾t crest 14693   topGenctg 14710   TopBasesctb 19267   2ndcc2ndc 19807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-fin 7532  df-fi 7883  df-card 8332  df-acn 8335  df-rest 14695  df-topgen 14716  df-bases 19270  df-2ndc 19809
This theorem is referenced by: (None)
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