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Theorem tx2ndc 20020
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 19815 . 2  |-  ( R  e.  2ndc  <->  E. r  e.  TopBases  ( r  ~<_  om  /\  ( topGen `
 r )  =  R ) )
2 is2ndc 19815 . 2  |-  ( S  e.  2ndc  <->  E. s  e.  TopBases  ( s  ~<_  om  /\  ( topGen `
 s )  =  S ) )
3 reeanv 3034 . . 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 822 . . . . 5  |-  ( ( ( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  (
s  ~<_  om  /\  ( topGen `
 s )  =  S ) )  <->  ( (
r  ~<_  om  /\  s  ~<_  om )  /\  (
( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S ) ) )
5 txbasval 19975 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( r 
tX  s ) )
6 eqid 2467 . . . . . . . . . . 11  |-  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
76txval 19933 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( r  tX  s )  =  (
topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
85, 7eqtrd 2508 . . . . . . . . 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 19936 . . . . . . . . . 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 8075 . . . . . . . . . . . 12  |-  om  e.  On
13 vex 3121 . . . . . . . . . . . . . . . 16  |-  s  e. 
_V
1413xpdom1 7628 . . . . . . . . . . . . . . 15  |-  ( r  ~<_  om  ->  ( r  X.  s )  ~<_  ( om 
X.  s ) )
15 omex 8072 . . . . . . . . . . . . . . . 16  |-  om  e.  _V
1615xpdom2 7624 . . . . . . . . . . . . . . 15  |-  ( s  ~<_  om  ->  ( om  X.  s )  ~<_  ( om 
X.  om ) )
17 domtr 7580 . . . . . . . . . . . . . . 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 8405 . . . . . . . . . . . . 13  |-  ( om 
X.  om )  ~~  om
21 domentr 7586 . . . . . . . . . . . . 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 8430 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
26 vex 3121 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
27 vex 3121 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
2826, 27xpex 6599 . . . . . . . . . . . . . 14  |-  ( x  X.  y )  e. 
_V
2925, 28fnmpt2i 6864 . . . . . . . . . . . . 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 5807 . . . . . . . . . . . 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 8450 . . . . . . . . . . 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 7580 . . . . . . . . . 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 19817 . . . . . . . . 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 2555 . . . . . . 7  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  e.  2ndc )
40 oveq12 6304 . . . . . . . 8  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  =  ( R 
tX  S ) )
4140eleq1d 2536 . . . . . . 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 2964 . . 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 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   Oncon0 4884    X. cxp 5003   dom cdm 5005   ran crn 5006    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   omcom 6695    ~~ cen 7525    ~<_ cdom 7526   cardccrd 8328   topGenctg 14710   TopBasesctb 19267   2ndcc2ndc 19807    tX ctx 19929
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  ax-inf2 8070
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-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-oi 7947  df-card 8332  df-acn 8335  df-topgen 14716  df-bases 19270  df-2ndc 19809  df-tx 19931
This theorem is referenced by: (None)
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