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Theorem tx2ndc 17675
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 17501 . 2  |-  ( R  e.  2ndc  <->  E. r  e.  TopBases  ( r  ~<_  om  /\  ( topGen `
 r )  =  R ) )
2 is2ndc 17501 . 2  |-  ( S  e.  2ndc  <->  E. s  e.  TopBases  ( s  ~<_  om  /\  ( topGen `
 s )  =  S ) )
3 reeanv 2867 . . 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 798 . . . . 5  |-  ( ( ( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  (
s  ~<_  om  /\  ( topGen `
 s )  =  S ) )  <->  ( (
r  ~<_  om  /\  s  ~<_  om )  /\  (
( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S ) ) )
5 txbasval 17630 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( r 
tX  s ) )
6 eqid 2435 . . . . . . . . . . 11  |-  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
76txval 17588 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( r  tX  s )  =  (
topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
85, 7eqtrd 2467 . . . . . . . . 9  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( topGen ` 
ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
98adantr 452 . . . . . . . 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 17591 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  e.  TopBases )
1110adantr 452 . . . . . . . . 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 7593 . . . . . . . . . . . 12  |-  om  e.  On
13 vex 2951 . . . . . . . . . . . . . . . 16  |-  s  e. 
_V
1413xpdom1 7199 . . . . . . . . . . . . . . 15  |-  ( r  ~<_  om  ->  ( r  X.  s )  ~<_  ( om 
X.  s ) )
15 omex 7590 . . . . . . . . . . . . . . . 16  |-  om  e.  _V
1615xpdom2 7195 . . . . . . . . . . . . . . 15  |-  ( s  ~<_  om  ->  ( om  X.  s )  ~<_  ( om 
X.  om ) )
17 domtr 7152 . . . . . . . . . . . . . . 15  |-  ( ( ( r  X.  s
)  ~<_  ( om  X.  s )  /\  ( om  X.  s )  ~<_  ( om  X.  om )
)  ->  ( r  X.  s )  ~<_  ( om 
X.  om ) )
1814, 16, 17syl2an 464 . . . . . . . . . . . . . 14  |-  ( ( r  ~<_  om  /\  s  ~<_  om )  ->  ( r  X.  s )  ~<_  ( om  X.  om )
)
1918adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  ~<_  ( om  X.  om ) )
20 xpomen 7889 . . . . . . . . . . . . 13  |-  ( om 
X.  om )  ~~  om
21 domentr 7158 . . . . . . . . . . . . 13  |-  ( ( ( r  X.  s
)  ~<_  ( om  X.  om )  /\  ( om  X.  om )  ~~  om )  ->  ( r  X.  s )  ~<_  om )
2219, 20, 21sylancl 644 . . . . . . . . . . . 12  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  ~<_  om )
23 ondomen 7910 . . . . . . . . . . . 12  |-  ( ( om  e.  On  /\  ( r  X.  s
)  ~<_  om )  ->  (
r  X.  s )  e.  dom  card )
2412, 22, 23sylancr 645 . . . . . . . . . . 11  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
r  X.  s )  e.  dom  card )
25 eqid 2435 . . . . . . . . . . . . . 14  |-  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
26 vex 2951 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
27 vex 2951 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
2826, 27xpex 4982 . . . . . . . . . . . . . 14  |-  ( x  X.  y )  e. 
_V
2925, 28fnmpt2i 6412 . . . . . . . . . . . . 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 5651 . . . . . . . . . . . 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 189 . . . . . . . . . . 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 7930 . . . . . . . . . . 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 58 . . . . . . . . . 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 7152 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y
) )  ~<_  om )
37 2ndci 17503 . . . . . . . . 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 643 . . . . . . . 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 2509 . . . . . . 7  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  e.  2ndc )
40 oveq12 6082 . . . . . . . 8  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  =  ( R 
tX  S ) )
4140eleq1d 2501 . . . . . . 7  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( ( topGen `  r
)  tX  ( topGen `  s ) )  e. 
2ndc 
<->  ( R  tX  S
)  e.  2ndc )
)
4239, 41syl5ibcom 212 . . . . . 6  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( ( topGen `  r
)  =  R  /\  ( topGen `  s )  =  S )  ->  ( R  tX  S )  e. 
2ndc ) )
4342expimpd 587 . . . . 5  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( (
( r  ~<_  om  /\  s  ~<_  om )  /\  (
( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S ) )  -> 
( R  tX  S
)  e.  2ndc )
)
444, 43syl5bi 209 . . . 4  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( (
( r  ~<_  om  /\  ( topGen `  r )  =  R )  /\  (
s  ~<_  om  /\  ( topGen `
 s )  =  S ) )  -> 
( R  tX  S
)  e.  2ndc )
)
4544rexlimivv 2827 . . 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 205 . 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 466 1  |-  ( ( R  e.  2ndc  /\  S  e.  2ndc )  ->  ( R  tX  S )  e. 
2ndc )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   Oncon0 4573   omcom 4837    X. cxp 4868   dom cdm 4870   ran crn 4871    Fn wfn 5441   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075    ~~ cen 7098    ~<_ cdom 7099   cardccrd 7814   topGenctg 13657   TopBasesctb 16954   2ndcc2ndc 17493    tX ctx 17584
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-acn 7821  df-topgen 13659  df-bases 16957  df-2ndc 17495  df-tx 17586
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