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Theorem tx2ndc 17605
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 17431 . 2  |-  ( R  e.  2ndc  <->  E. r  e.  TopBases  ( r  ~<_  om  /\  ( topGen `
 r )  =  R ) )
2 is2ndc 17431 . 2  |-  ( S  e.  2ndc  <->  E. s  e.  TopBases  ( s  ~<_  om  /\  ( topGen `
 s )  =  S ) )
3 reeanv 2819 . . 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 17560 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( ( topGen `
 r )  tX  ( topGen `  s )
)  =  ( r 
tX  s ) )
6 eqid 2388 . . . . . . . . . . 11  |-  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ran  (
x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
76txval 17518 . . . . . . . . . 10  |-  ( ( r  e.  TopBases  /\  s  e. 
TopBases )  ->  ( r  tX  s )  =  (
topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) ) ) )
85, 7eqtrd 2420 . . . . . . . . 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 17521 . . . . . . . . . 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 7535 . . . . . . . . . . . 12  |-  om  e.  On
13 vex 2903 . . . . . . . . . . . . . . . 16  |-  s  e. 
_V
1413xpdom1 7144 . . . . . . . . . . . . . . 15  |-  ( r  ~<_  om  ->  ( r  X.  s )  ~<_  ( om 
X.  s ) )
15 omex 7532 . . . . . . . . . . . . . . . 16  |-  om  e.  _V
1615xpdom2 7140 . . . . . . . . . . . . . . 15  |-  ( s  ~<_  om  ->  ( om  X.  s )  ~<_  ( om 
X.  om ) )
17 domtr 7097 . . . . . . . . . . . . . . 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 7831 . . . . . . . . . . . . 13  |-  ( om 
X.  om )  ~~  om
21 domentr 7103 . . . . . . . . . . . . 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 7852 . . . . . . . . . . . 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 2388 . . . . . . . . . . . . . 14  |-  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )  =  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y ) )
26 vex 2903 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
27 vex 2903 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
2826, 27xpex 4931 . . . . . . . . . . . . . 14  |-  ( x  X.  y )  e. 
_V
2925, 28fnmpt2i 6360 . . . . . . . . . . . . 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 5600 . . . . . . . . . . . 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 7872 . . . . . . . . . . 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 7097 . . . . . . . . . 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 17433 . . . . . . . . 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 2462 . . . . . . 7  |-  ( ( ( r  e.  TopBases  /\  s  e.  TopBases )  /\  (
r  ~<_  om  /\  s  ~<_  om ) )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  e.  2ndc )
40 oveq12 6030 . . . . . . . 8  |-  ( ( ( topGen `  r )  =  R  /\  ( topGen `
 s )  =  S )  ->  (
( topGen `  r )  tX  ( topGen `  s )
)  =  ( R 
tX  S ) )
4140eleq1d 2454 . . . . . . 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 2779 . . 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 1649    e. wcel 1717   E.wrex 2651   class class class wbr 4154   Oncon0 4523   omcom 4786    X. cxp 4817   dom cdm 4819   ran crn 4820    Fn wfn 5390   -onto->wfo 5393   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023    ~~ cen 7043    ~<_ cdom 7044   cardccrd 7756   topGenctg 13593   TopBasesctb 16886   2ndcc2ndc 17423    tX ctx 17514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-oi 7413  df-card 7760  df-acn 7763  df-topgen 13595  df-bases 16889  df-2ndc 17425  df-tx 17516
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