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Theorem txopn 12269
Description: The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
txopn  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ( R 
tX  S ) )

Proof of Theorem txopn
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2113 . . . . . 6  |-  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  =  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
21txbasex 12261 . . . . 5  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  e. 
_V )
3 bastg 12066 . . . . 5  |-  ( ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) )  e.  _V  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  C_  ( topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
42, 3syl 14 . . . 4  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  C_  ( topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
54adantr 272 . . 3  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) )  C_  ( topGen `
 ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
6 eqid 2113 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 4511 . . . . . . . 8  |-  ( u  =  A  ->  (
u  X.  v )  =  ( A  X.  v ) )
87eqeq2d 2124 . . . . . . 7  |-  ( u  =  A  ->  (
( A  X.  B
)  =  ( u  X.  v )  <->  ( A  X.  B )  =  ( A  X.  v ) ) )
9 xpeq2 4512 . . . . . . . 8  |-  ( v  =  B  ->  ( A  X.  v )  =  ( A  X.  B
) )
109eqeq2d 2124 . . . . . . 7  |-  ( v  =  B  ->  (
( A  X.  B
)  =  ( A  X.  v )  <->  ( A  X.  B )  =  ( A  X.  B ) ) )
118, 10rspc2ev 2772 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S  /\  ( A  X.  B
)  =  ( A  X.  B ) )  ->  E. u  e.  R  E. v  e.  S  ( A  X.  B
)  =  ( u  X.  v ) )
126, 11mp3an3 1285 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  ->  E. u  e.  R  E. v  e.  S  ( A  X.  B
)  =  ( u  X.  v ) )
13 xpexg 4611 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2113 . . . . . . 7  |-  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  =  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )
1514elrnmpog 5835 . . . . . 6  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  B
)  e.  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  <->  E. u  e.  R  E. v  e.  S  ( A  X.  B
)  =  ( u  X.  v ) ) )
1613, 15syl 14 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A  X.  B )  e.  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) )  <->  E. u  e.  R  E. v  e.  S  ( A  X.  B )  =  ( u  X.  v ) ) )
1712, 16mpbird 166 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A  X.  B
)  e.  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) ) )
1817adantl 273 . . 3  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) ) )
195, 18sseldd 3062 . 2  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ( topGen ` 
ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
201txval 12259 . . 3  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S
)  =  ( topGen ` 
ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
2120adantr 272 . 2  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( R  tX  S
)  =  ( topGen ` 
ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
2219, 21eleqtrrd 2192 1  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ( R 
tX  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1312    e. wcel 1461   E.wrex 2389   _Vcvv 2655    C_ wss 3035    X. cxp 4495   ran crn 4498   ` cfv 5079  (class class class)co 5726    e. cmpo 5728   topGenctg 11971    tX ctx 12256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5989  df-2nd 5990  df-topgen 11977  df-tx 12257
This theorem is referenced by:  txbasval  12271  neitx  12272  tx1cn  12273  tx2cn  12274  txlm  12283
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