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Theorem txdis 13444
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 13252 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 distop 13252 . . . . 5  |-  ( B  e.  W  ->  ~P B  e.  Top )
3 unipw 4214 . . . . . . 7  |-  U. ~P A  =  A
43eqcomi 2181 . . . . . 6  |-  A  = 
U. ~P A
5 unipw 4214 . . . . . . 7  |-  U. ~P B  =  B
65eqcomi 2181 . . . . . 6  |-  B  = 
U. ~P B
74, 6txuni 13430 . . . . 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 3210 . . . 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 3968 . . 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 3586 . . . . . . . . 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 6160 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 1st `  y )  e.  A )
16 snelpwi 4209 . . . . . . . 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 6161 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 2nd `  y )  e.  B )
19 snelpwi 4209 . . . . . . . 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 3623 . . . . . . . 8  |-  y  e. 
{ y }
22 1st2nd2 6170 . . . . . . . . . 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 3604 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { y }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
2521, 24eleqtrid 2266 . . . . . . 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 2255 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  x
)
2827snssd 3736 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  C_  x
)
29 xpeq1 4637 . . . . . . . . . 10  |-  ( z  =  { ( 1st `  y ) }  ->  ( z  X.  w )  =  ( { ( 1st `  y ) }  X.  w ) )
3029eleq2d 2247 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( y  e.  ( z  X.  w )  <->  y  e.  ( { ( 1st `  y
) }  X.  w
) ) )
3129sseq1d 3184 . . . . . . . . 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 4638 . . . . . . . . . . 11  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } ) )
34 1stexg 6162 . . . . . . . . . . . . 13  |-  ( y  e.  _V  ->  ( 1st `  y )  e. 
_V )
3534elv 2741 . . . . . . . . . . . 12  |-  ( 1st `  y )  e.  _V
36 2ndexg 6163 . . . . . . . . . . . . 13  |-  ( y  e.  _V  ->  ( 2nd `  y )  e. 
_V )
3736elv 2741 . . . . . . . . . . . 12  |-  ( 2nd `  y )  e.  _V
3835, 37xpsn 5688 . . . . . . . . . . 11  |-  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } )  =  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }
3933, 38eqtrdi 2226 . . . . . . . . . 10  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
4039eleq2d 2247 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( y  e.  ( { ( 1st `  y
) }  X.  w
)  <->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } ) )
4139sseq1d 3184 . . . . . . . . 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 2856 . . . . . . 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 2557 . . . 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 13426 . . . . 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 3161 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  X.  B )  C_  ( ~P A  tX  ~P B
) )
5112, 50eqssd 3172 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 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737    C_ wss 3129   ~Pcpw 3574   {csn 3591   <.cop 3594   U.cuni 3807    X. cxp 4621   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Topctop 13162    tX ctx 13419
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-topgen 12657  df-top 13163  df-topon 13176  df-bases 13208  df-tx 13420
This theorem is referenced by: (None)
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