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Theorem 2ndcrest 17517
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 17509 . . 3  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  x  e. 
TopBases )
3 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  A  e.  V )
4 tgrest 17223 . . . . . . . 8  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
52, 3, 4syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
6 restbas 17222 . . . . . . . . 9  |-  ( x  e.  TopBases  ->  ( xt  A )  e.  TopBases )
76ad2antlr 708 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  e.  TopBases )
8 restval 13654 . . . . . . . . . 10  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
92, 3, 8syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
10 1stcrestlem 17515 . . . . . . . . . 10  |-  ( x  ~<_  om  ->  ran  ( y  e.  x  |->  ( y  i^i  A ) )  ~<_  om )
1110adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ran  ( y  e.  x  |->  ( y  i^i  A
) )  ~<_  om )
129, 11eqbrtrd 4232 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  ~<_  om )
13 2ndci 17511 . . . . . . . 8  |-  ( ( ( xt  A )  e.  TopBases  /\  ( xt  A )  ~<_  om )  ->  ( topGen `  ( xt  A
) )  e.  2ndc )
147, 12, 13syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  e.  2ndc )
155, 14eqeltrrd 2511 . . . . . 6  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )t  A
)  e.  2ndc )
16 oveq1 6088 . . . . . . 7  |-  ( (
topGen `  x )  =  J  ->  ( ( topGen `
 x )t  A )  =  ( Jt  A ) )
1716eleq1d 2502 . . . . . 6  |-  ( (
topGen `  x )  =  J  ->  ( (
( topGen `  x )t  A
)  e.  2ndc  <->  ( Jt  A
)  e.  2ndc )
)
1815, 17syl5ibcom 212 . . . . 5  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )  =  J  ->  ( Jt  A )  e.  2ndc )
)
1918expimpd 587 . . . 4  |-  ( ( A  e.  V  /\  x  e.  TopBases )  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  ( Jt  A )  e.  2ndc ) )
2019rexlimdva 2830 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  ( Jt  A
)  e.  2ndc )
)
211, 20syl5bi 209 . 2  |-  ( A  e.  V  ->  ( J  e.  2ndc  ->  ( Jt  A )  e.  2ndc ) )
2221impcom 420 1  |-  ( ( J  e.  2ndc  /\  A  e.  V )  ->  ( Jt  A )  e.  2ndc )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319   class class class wbr 4212    e. cmpt 4266   omcom 4845   ran crn 4879   ` cfv 5454  (class class class)co 6081    ~<_ cdom 7107   ↾t crest 13648   topGenctg 13665   TopBasesctb 16962   2ndcc2ndc 17501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-fin 7113  df-fi 7416  df-card 7826  df-acn 7829  df-rest 13650  df-topgen 13667  df-bases 16965  df-2ndc 17503
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