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Theorem txdis 12476
Description: The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txdis  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  =  ~P ( A  X.  B
) )

Proof of Theorem txdis
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 12284 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 distop 12284 . . . . 5  |-  ( B  e.  W  ->  ~P B  e.  Top )
3 unipw 4143 . . . . . . 7  |-  U. ~P A  =  A
43eqcomi 2144 . . . . . 6  |-  A  = 
U. ~P A
5 unipw 4143 . . . . . . 7  |-  U. ~P B  =  B
65eqcomi 2144 . . . . . 6  |-  B  = 
U. ~P B
74, 6txuni 12462 . . . . 5  |-  ( ( ~P A  e.  Top  /\ 
~P B  e.  Top )  ->  ( A  X.  B )  =  U. ( ~P A  tX  ~P B ) )
81, 2, 7syl2an 287 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  =  U. ( ~P A  tX  ~P B
) )
9 eqimss2 3153 . . . 4  |-  ( ( A  X.  B )  =  U. ( ~P A  tX  ~P B
)  ->  U. ( ~P A  tX  ~P B
)  C_  ( A  X.  B ) )
108, 9syl 14 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. ( ~P A  tX 
~P B )  C_  ( A  X.  B
) )
11 sspwuni 3901 . . 3  |-  ( ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
)  <->  U. ( ~P A  tX 
~P B )  C_  ( A  X.  B
) )
1210, 11sylibr 133 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
) )
13 elelpwi 3523 . . . . . . . . 9  |-  ( ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) )  ->  y  e.  ( A  X.  B ) )
1413adantl 275 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  ( A  X.  B ) )
15 xp1st 6067 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 1st `  y )  e.  A )
16 snelpwi 4138 . . . . . . . 8  |-  ( ( 1st `  y )  e.  A  ->  { ( 1st `  y ) }  e.  ~P A
)
1714, 15, 163syl 17 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { ( 1st `  y ) }  e.  ~P A )
18 xp2nd 6068 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 2nd `  y )  e.  B )
19 snelpwi 4138 . . . . . . . 8  |-  ( ( 2nd `  y )  e.  B  ->  { ( 2nd `  y ) }  e.  ~P B
)
2014, 18, 193syl 17 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { ( 2nd `  y ) }  e.  ~P B )
21 vsnid 3560 . . . . . . . 8  |-  y  e. 
{ y }
22 1st2nd2 6077 . . . . . . . . . 10  |-  ( y  e.  ( A  X.  B )  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2314, 22syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2423sneqd 3541 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { y }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
2521, 24eleqtrid 2229 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
26 simprl 521 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  x
)
2723, 26eqeltrrd 2218 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  x
)
2827snssd 3669 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  C_  x
)
29 xpeq1 4557 . . . . . . . . . 10  |-  ( z  =  { ( 1st `  y ) }  ->  ( z  X.  w )  =  ( { ( 1st `  y ) }  X.  w ) )
3029eleq2d 2210 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( y  e.  ( z  X.  w )  <->  y  e.  ( { ( 1st `  y
) }  X.  w
) ) )
3129sseq1d 3127 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( ( z  X.  w
)  C_  x  <->  ( {
( 1st `  y
) }  X.  w
)  C_  x )
)
3230, 31anbi12d 465 . . . . . . . 8  |-  ( z  =  { ( 1st `  y ) }  ->  ( ( y  e.  ( z  X.  w )  /\  ( z  X.  w )  C_  x
)  <->  ( y  e.  ( { ( 1st `  y ) }  X.  w )  /\  ( { ( 1st `  y
) }  X.  w
)  C_  x )
) )
33 xpeq2 4558 . . . . . . . . . . 11  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } ) )
34 1stexg 6069 . . . . . . . . . . . . 13  |-  ( y  e.  _V  ->  ( 1st `  y )  e. 
_V )
3534elv 2691 . . . . . . . . . . . 12  |-  ( 1st `  y )  e.  _V
36 2ndexg 6070 . . . . . . . . . . . . 13  |-  ( y  e.  _V  ->  ( 2nd `  y )  e. 
_V )
3736elv 2691 . . . . . . . . . . . 12  |-  ( 2nd `  y )  e.  _V
3835, 37xpsn 5600 . . . . . . . . . . 11  |-  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } )  =  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }
3933, 38eqtrdi 2189 . . . . . . . . . 10  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
4039eleq2d 2210 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( y  e.  ( { ( 1st `  y
) }  X.  w
)  <->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } ) )
4139sseq1d 3127 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( ( { ( 1st `  y ) }  X.  w )  C_  x  <->  {
<. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) )
4240, 41anbi12d 465 . . . . . . . 8  |-  ( w  =  { ( 2nd `  y ) }  ->  ( ( y  e.  ( { ( 1st `  y
) }  X.  w
)  /\  ( {
( 1st `  y
) }  X.  w
)  C_  x )  <->  ( y  e.  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  /\  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) ) )
4332, 42rspc2ev 2805 . . . . . . 7  |-  ( ( { ( 1st `  y
) }  e.  ~P A  /\  { ( 2nd `  y ) }  e.  ~P B  /\  (
y  e.  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  /\  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) )  ->  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )
4417, 20, 25, 28, 43syl112anc 1221 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )
4544expr 373 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  y  e.  x )  ->  (
x  e.  ~P ( A  X.  B )  ->  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) ) )
4645ralrimdva 2513 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ~P ( A  X.  B
)  ->  A. y  e.  x  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
47 eltx 12458 . . . . 5  |-  ( ( ~P A  e.  Top  /\ 
~P B  e.  Top )  ->  ( x  e.  ( ~P A  tX  ~P B )  <->  A. y  e.  x  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
481, 2, 47syl2an 287 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ( ~P A  tX  ~P B )  <->  A. y  e.  x  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
4946, 48sylibrd 168 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ~P ( A  X.  B
)  ->  x  e.  ( ~P A  tX  ~P B ) ) )
5049ssrdv 3104 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  X.  B )  C_  ( ~P A  tX  ~P B
) )
5112, 50eqssd 3115 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  =  ~P ( A  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2687    C_ wss 3072   ~Pcpw 3511   {csn 3528   <.cop 3531   U.cuni 3740    X. cxp 4541   ` cfv 5127  (class class class)co 5778   1stc1st 6040   2ndc2nd 6041   Topctop 12194    tX ctx 12451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-topgen 12171  df-top 12195  df-topon 12208  df-bases 12240  df-tx 12452
This theorem is referenced by: (None)
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