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Theorem tx2ndc 19224
Description: The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
tx2ndc  |-  ( ( R  e.  2ndc  /\  S  e.  2ndc )  ->  ( R  tX  S )  e. 
2ndc )

Proof of Theorem tx2ndc
Dummy variables  s 
r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2ndc 19050 . 2  |-  ( R  e.  2ndc  <->  E. r  e.  TopBases  ( r  ~<_  om  /\  ( topGen `
 r )  =  R ) )
2 is2ndc 19050 . 2  |-  ( S  e.  2ndc  <->  E. s  e.  TopBases  ( s  ~<_  om  /\  ( topGen `
 s )  =  S ) )
3 reeanv 2888 . . 3  |-  ( E. r  e.  TopBases  E. s  e. 
TopBases  ( ( r  ~<_  om 
/\  ( topGen `  r
)  =  R )  /\  ( s  ~<_  om 
/\  ( topGen `  s
)  =  S ) )  <->  ( E. r  e. 
TopBases  ( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  E. s  e.  TopBases  ( s  ~<_  om  /\  ( topGen `  s
)  =  S ) ) )
4 an4 820 . . . . 5  |-  ( ( ( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  (
s  ~<_  om  /\  ( topGen `
 s )  =  S ) )  <->  ( (
r  ~<_  om  /\  s  ~<_  om )  /\  (
( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S ) ) )
5 txbasval 19179 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( r 
tX  s ) )
6 eqid 2443 . . . . . . . . . . 11  |-  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
76txval 19137 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( r  tX  s )  =  (
topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
85, 7eqtrd 2475 . . . . . . . . 9  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( topGen ` 
ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
98adantr 465 . . . . . . . 8  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  =  ( topGen ` 
ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
106txbas 19140 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  e.  TopBases )
1110adantr 465 . . . . . . . . 9  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y
) )  e.  TopBases )
12 omelon 7852 . . . . . . . . . . . 12  |-  om  e.  On
13 vex 2975 . . . . . . . . . . . . . . . 16  |-  s  e. 
_V
1413xpdom1 7410 . . . . . . . . . . . . . . 15  |-  ( r  ~<_  om  ->  ( r  X.  s )  ~<_  ( om 
X.  s ) )
15 omex 7849 . . . . . . . . . . . . . . . 16  |-  om  e.  _V
1615xpdom2 7406 . . . . . . . . . . . . . . 15  |-  ( s  ~<_  om  ->  ( om  X.  s )  ~<_  ( om 
X.  om ) )
17 domtr 7362 . . . . . . . . . . . . . . 15  |-  ( ( ( r  X.  s
)  ~<_  ( om  X.  s )  /\  ( om  X.  s )  ~<_  ( om  X.  om )
)  ->  ( r  X.  s )  ~<_  ( om 
X.  om ) )
1814, 16, 17syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( r  ~<_  om  /\  s  ~<_  om )  ->  ( r  X.  s )  ~<_  ( om  X.  om )
)
1918adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  ~<_  ( om  X.  om ) )
20 xpomen 8182 . . . . . . . . . . . . 13  |-  ( om 
X.  om )  ~~  om
21 domentr 7368 . . . . . . . . . . . . 13  |-  ( ( ( r  X.  s
)  ~<_  ( om  X.  om )  /\  ( om  X.  om )  ~~  om )  ->  ( r  X.  s )  ~<_  om )
2219, 20, 21sylancl 662 . . . . . . . . . . . 12  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  ~<_  om )
23 ondomen 8207 . . . . . . . . . . . 12  |-  ( ( om  e.  On  /\  ( r  X.  s
)  ~<_  om )  ->  (
r  X.  s )  e.  dom  card )
2412, 22, 23sylancr 663 . . . . . . . . . . 11  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  e.  dom  card )
25 eqid 2443 . . . . . . . . . . . . . 14  |-  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
26 vex 2975 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
27 vex 2975 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
2826, 27xpex 6508 . . . . . . . . . . . . . 14  |-  ( x  X.  y )  e. 
_V
2925, 28fnmpt2i 6643 . . . . . . . . . . . . 13  |-  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  Fn  ( r  X.  s )
3029a1i 11 . . . . . . . . . . . 12  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  Fn  ( r  X.  s ) )
31 dffn4 5626 . . . . . . . . . . . 12  |-  ( ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  Fn  ( r  X.  s )  <->  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) : ( r  X.  s
) -onto-> ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )
3230, 31sylib 196 . . . . . . . . . . 11  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) : ( r  X.  s ) -onto-> ran  ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) ) )
33 fodomnum 8227 . . . . . . . . . . 11  |-  ( ( r  X.  s )  e.  dom  card  ->  ( ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) ) : ( r  X.  s )
-onto->
ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  ->  ran  ( x  e.  r ,  y  e.  s 
|->  ( x  X.  y
) )  ~<_  ( r  X.  s ) ) )
3424, 32, 33sylc 60 . . . . . . . . . 10  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y
) )  ~<_  ( r  X.  s ) )
35 domtr 7362 . . . . . . . . . 10  |-  ( ( ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  ~<_  ( r  X.  s )  /\  ( r  X.  s )  ~<_  om )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  ~<_  om )
3634, 22, 35syl2anc 661 . . . . . . . . 9  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y
) )  ~<_  om )
37 2ndci 19052 . . . . . . . . 9  |-  ( ( ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  e.  TopBases 
/\  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  ~<_  om )  ->  ( topGen ` 
ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )  e.  2ndc )
3811, 36, 37syl2anc 661 . . . . . . . 8  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  ( topGen `
 ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) )  e.  2ndc )
399, 38eqeltrd 2517 . . . . . . 7  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  e.  2ndc )
40 oveq12 6100 . . . . . . . 8  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  =  ( R 
tX  S ) )
4140eleq1d 2509 . . . . . . 7  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( ( topGen `  r
)  tX  ( topGen `  s ) )  e. 
2ndc 
<->  ( R  tX  S
)  e.  2ndc )
)
4239, 41syl5ibcom 220 . . . . . 6  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( ( topGen `  r
)  =  R  /\  ( topGen `  s )  =  S )  ->  ( R  tX  S )  e. 
2ndc ) )
4342expimpd 603 . . . . 5  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( (
( r  ~<_  om  /\  s  ~<_  om )  /\  (
( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S ) )  -> 
( R  tX  S
)  e.  2ndc )
)
444, 43syl5bi 217 . . . 4  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( (
( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  (
s  ~<_  om  /\  ( topGen `
 s )  =  S ) )  -> 
( R  tX  S
)  e.  2ndc )
)
4544rexlimivv 2846 . . 3  |-  ( E. r  e.  TopBases  E. s  e. 
TopBases  ( ( r  ~<_  om 
/\  ( topGen `  r
)  =  R )  /\  ( s  ~<_  om 
/\  ( topGen `  s
)  =  S ) )  ->  ( R  tX  S )  e.  2ndc )
463, 45sylbir 213 . 2  |-  ( ( E. r  e.  TopBases  ( r  ~<_  om  /\  ( topGen `
 r )  =  R )  /\  E. s  e.  TopBases  ( s  ~<_  om  /\  ( topGen `  s
)  =  S ) )  ->  ( R  tX  S )  e.  2ndc )
471, 2, 46syl2anb 479 1  |-  ( ( R  e.  2ndc  /\  S  e.  2ndc )  ->  ( R  tX  S )  e. 
2ndc )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   class class class wbr 4292   Oncon0 4719    X. cxp 4838   dom cdm 4840   ran crn 4841    Fn wfn 5413   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   omcom 6476    ~~ cen 7307    ~<_ cdom 7308   cardccrd 8105   topGenctg 14376   TopBasesctb 18502   2ndcc2ndc 19042    tX ctx 19133
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  ax-inf2 7847
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-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-oi 7724  df-card 8109  df-acn 8112  df-topgen 14382  df-bases 18505  df-2ndc 19044  df-tx 19135
This theorem is referenced by: (None)
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