MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tx2ndc Unicode version

Theorem tx2ndc 17345
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 17172 . 2  |-  ( R  e.  2ndc  <->  E. r  e.  TopBases  ( r  ~<_  om  /\  ( topGen `
 r )  =  R ) )
2 is2ndc 17172 . 2  |-  ( S  e.  2ndc  <->  E. s  e.  TopBases  ( s  ~<_  om  /\  ( topGen `
 s )  =  S ) )
3 reeanv 2707 . . 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 797 . . . . 5  |-  ( ( ( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  (
s  ~<_  om  /\  ( topGen `
 s )  =  S ) )  <->  ( (
r  ~<_  om  /\  s  ~<_  om )  /\  (
( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S ) ) )
5 txbasval 17301 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( r 
tX  s ) )
6 eqid 2283 . . . . . . . . . . 11  |-  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
76txval 17259 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( r  tX  s )  =  (
topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
85, 7eqtrd 2315 . . . . . . . . 9  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( topGen ` 
ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
98adantr 451 . . . . . . . 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 17262 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  e.  TopBases )
1110adantr 451 . . . . . . . . 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 7347 . . . . . . . . . . . 12  |-  om  e.  On
13 vex 2791 . . . . . . . . . . . . . . . 16  |-  s  e. 
_V
1413xpdom1 6961 . . . . . . . . . . . . . . 15  |-  ( r  ~<_  om  ->  ( r  X.  s )  ~<_  ( om 
X.  s ) )
15 omex 7344 . . . . . . . . . . . . . . . 16  |-  om  e.  _V
1615xpdom2 6957 . . . . . . . . . . . . . . 15  |-  ( s  ~<_  om  ->  ( om  X.  s )  ~<_  ( om 
X.  om ) )
17 domtr 6914 . . . . . . . . . . . . . . 15  |-  ( ( ( r  X.  s
)  ~<_  ( om  X.  s )  /\  ( om  X.  s )  ~<_  ( om  X.  om )
)  ->  ( r  X.  s )  ~<_  ( om 
X.  om ) )
1814, 16, 17syl2an 463 . . . . . . . . . . . . . 14  |-  ( ( r  ~<_  om  /\  s  ~<_  om )  ->  ( r  X.  s )  ~<_  ( om  X.  om )
)
1918adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  ~<_  ( om  X.  om ) )
20 xpomen 7643 . . . . . . . . . . . . 13  |-  ( om 
X.  om )  ~~  om
21 domentr 6920 . . . . . . . . . . . . 13  |-  ( ( ( r  X.  s
)  ~<_  ( om  X.  om )  /\  ( om  X.  om )  ~~  om )  ->  ( r  X.  s )  ~<_  om )
2219, 20, 21sylancl 643 . . . . . . . . . . . 12  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  ~<_  om )
23 ondomen 7664 . . . . . . . . . . . 12  |-  ( ( om  e.  On  /\  ( r  X.  s
)  ~<_  om )  ->  (
r  X.  s )  e.  dom  card )
2412, 22, 23sylancr 644 . . . . . . . . . . 11  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  e.  dom  card )
25 eqid 2283 . . . . . . . . . . . . . 14  |-  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
26 vex 2791 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
27 vex 2791 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
2826, 27xpex 4801 . . . . . . . . . . . . . 14  |-  ( x  X.  y )  e. 
_V
2925, 28fnmpt2i 6193 . . . . . . . . . . . . 13  |-  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  Fn  ( r  X.  s )
3029a1i 10 . . . . . . . . . . . 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 5457 . . . . . . . . . . . 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 188 . . . . . . . . . . 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 7684 . . . . . . . . . . 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 56 . . . . . . . . . 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 6914 . . . . . . . . . 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 642 . . . . . . . . 9  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y
) )  ~<_  om )
37 2ndci 17174 . . . . . . . . 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 642 . . . . . . . 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 2357 . . . . . . 7  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  e.  2ndc )
40 oveq12 5867 . . . . . . . 8  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  =  ( R 
tX  S ) )
4140eleq1d 2349 . . . . . . 7  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( ( topGen `  r
)  tX  ( topGen `  s ) )  e. 
2ndc 
<->  ( R  tX  S
)  e.  2ndc )
)
4239, 41syl5ibcom 211 . . . . . 6  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( ( topGen `  r
)  =  R  /\  ( topGen `  s )  =  S )  ->  ( R  tX  S )  e. 
2ndc ) )
4342expimpd 586 . . . . 5  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( (
( r  ~<_  om  /\  s  ~<_  om )  /\  (
( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S ) )  -> 
( R  tX  S
)  e.  2ndc )
)
444, 43syl5bi 208 . . . 4  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( (
( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  (
s  ~<_  om  /\  ( topGen `
 s )  =  S ) )  -> 
( R  tX  S
)  e.  2ndc )
)
4544rexlimivv 2672 . . 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 204 . 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 465 1  |-  ( ( R  e.  2ndc  /\  S  e.  2ndc )  ->  ( R  tX  S )  e. 
2ndc )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689   ran crn 4690    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ~~ cen 6860    ~<_ cdom 6861   cardccrd 7568   topGenctg 13342   TopBasesctb 16635   2ndcc2ndc 17164    tX ctx 17255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-acn 7575  df-topgen 13344  df-bases 16638  df-2ndc 17166  df-tx 17257
  Copyright terms: Public domain W3C validator