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Theorem txdis 13357
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 13165 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 distop 13165 . . . . 5  |-  ( B  e.  W  ->  ~P B  e.  Top )
3 unipw 4211 . . . . . . 7  |-  U. ~P A  =  A
43eqcomi 2179 . . . . . 6  |-  A  = 
U. ~P A
5 unipw 4211 . . . . . . 7  |-  U. ~P B  =  B
65eqcomi 2179 . . . . . 6  |-  B  = 
U. ~P B
74, 6txuni 13343 . . . . 5  |-  ( ( ~P A  e.  Top  /\ 
~P B  e.  Top )  ->  ( A  X.  B )  =  U. ( ~P A  tX  ~P B ) )
81, 2, 7syl2an 289 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  =  U. ( ~P A  tX  ~P B
) )
9 eqimss2 3208 . . . 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 3966 . . 3  |-  ( ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
)  <->  U. ( ~P A  tX 
~P B )  C_  ( A  X.  B
) )
1210, 11sylibr 134 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
) )
13 elelpwi 3584 . . . . . . . . 9  |-  ( ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) )  ->  y  e.  ( A  X.  B ) )
1413adantl 277 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  ( A  X.  B ) )
15 xp1st 6156 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 1st `  y )  e.  A )
16 snelpwi 4206 . . . . . . . 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 6157 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 2nd `  y )  e.  B )
19 snelpwi 4206 . . . . . . . 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 3621 . . . . . . . 8  |-  y  e. 
{ y }
22 1st2nd2 6166 . . . . . . . . . 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 3602 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { y }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
2521, 24eleqtrid 2264 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
26 simprl 529 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  x
)
2723, 26eqeltrrd 2253 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  x
)
2827snssd 3734 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  C_  x
)
29 xpeq1 4634 . . . . . . . . . 10  |-  ( z  =  { ( 1st `  y ) }  ->  ( z  X.  w )  =  ( { ( 1st `  y ) }  X.  w ) )
3029eleq2d 2245 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( y  e.  ( z  X.  w )  <->  y  e.  ( { ( 1st `  y
) }  X.  w
) ) )
3129sseq1d 3182 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( ( z  X.  w
)  C_  x  <->  ( {
( 1st `  y
) }  X.  w
)  C_  x )
)
3230, 31anbi12d 473 . . . . . . . 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 4635 . . . . . . . . . . 11  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } ) )
34 1stexg 6158 . . . . . . . . . . . . 13  |-  ( y  e.  _V  ->  ( 1st `  y )  e. 
_V )
3534elv 2739 . . . . . . . . . . . 12  |-  ( 1st `  y )  e.  _V
36 2ndexg 6159 . . . . . . . . . . . . 13  |-  ( y  e.  _V  ->  ( 2nd `  y )  e. 
_V )
3736elv 2739 . . . . . . . . . . . 12  |-  ( 2nd `  y )  e.  _V
3835, 37xpsn 5684 . . . . . . . . . . 11  |-  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } )  =  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }
3933, 38eqtrdi 2224 . . . . . . . . . 10  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
4039eleq2d 2245 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( y  e.  ( { ( 1st `  y
) }  X.  w
)  <->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } ) )
4139sseq1d 3182 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( ( { ( 1st `  y ) }  X.  w )  C_  x  <->  {
<. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) )
4240, 41anbi12d 473 . . . . . . . 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 2854 . . . . . . 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 1242 . . . . . 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 375 . . . . 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 2555 . . . 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 13339 . . . . 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 289 . . . 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 169 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ~P ( A  X.  B
)  ->  x  e.  ( ~P A  tX  ~P B ) ) )
5049ssrdv 3159 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  X.  B )  C_  ( ~P A  tX  ~P B
) )
5112, 50eqssd 3170 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 104    <-> wb 105    = wceq 1353    e. wcel 2146   A.wral 2453   E.wrex 2454   _Vcvv 2735    C_ wss 3127   ~Pcpw 3572   {csn 3589   <.cop 3592   U.cuni 3805    X. cxp 4618   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   Topctop 13075    tX ctx 13332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-topgen 12640  df-top 13076  df-topon 13089  df-bases 13121  df-tx 13333
This theorem is referenced by: (None)
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