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Theorem 2ndcrest 20121
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 20113 . . 3  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simplr 753 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  x  e. 
TopBases )
3 simpll 751 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  A  e.  V )
4 tgrest 19827 . . . . . . . 8  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
52, 3, 4syl2anc 659 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  =  ( (
topGen `  x )t  A ) )
6 restbas 19826 . . . . . . . . 9  |-  ( x  e.  TopBases  ->  ( xt  A )  e.  TopBases )
76ad2antlr 724 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  e.  TopBases )
8 restval 14916 . . . . . . . . . 10  |-  ( ( x  e.  TopBases  /\  A  e.  V )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
92, 3, 8syl2anc 659 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  =  ran  ( y  e.  x  |->  ( y  i^i  A
) ) )
10 1stcrestlem 20119 . . . . . . . . . 10  |-  ( x  ~<_  om  ->  ran  ( y  e.  x  |->  ( y  i^i  A ) )  ~<_  om )
1110adantl 464 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ran  ( y  e.  x  |->  ( y  i^i  A
) )  ~<_  om )
129, 11eqbrtrd 4459 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
xt 
A )  ~<_  om )
13 2ndci 20115 . . . . . . . 8  |-  ( ( ( xt  A )  e.  TopBases  /\  ( xt  A )  ~<_  om )  ->  ( topGen `  ( xt  A
) )  e.  2ndc )
147, 12, 13syl2anc 659 . . . . . . 7  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  ( topGen `
 ( xt  A ) )  e.  2ndc )
155, 14eqeltrrd 2543 . . . . . 6  |-  ( ( ( A  e.  V  /\  x  e.  TopBases )  /\  x  ~<_  om )  ->  (
( topGen `  x )t  A
)  e.  2ndc )
16 oveq1 6277 . . . . . . 7  |-  ( (
topGen `  x )  =  J  ->  ( ( topGen `
 x )t  A )  =  ( Jt  A ) )
1716eleq1d 2523 . . . . . 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 601 . . . 4  |-  ( ( A  e.  V  /\  x  e.  TopBases )  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  ( Jt  A )  e.  2ndc ) )
2019rexlimdva 2946 . . 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 428 1  |-  ( ( J  e.  2ndc  /\  A  e.  V )  ->  ( Jt  A )  e.  2ndc )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805    i^i cin 3460   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   ` cfv 5570  (class class class)co 6270   omcom 6673    ~<_ cdom 7507   ↾t crest 14910   topGenctg 14927   TopBasesctb 19565   2ndcc2ndc 20105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-fin 7513  df-fi 7863  df-card 8311  df-acn 8314  df-rest 14912  df-topgen 14933  df-bases 19568  df-2ndc 20107
This theorem is referenced by: (None)
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