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Theorem txss12 20272
Description: Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
txss12  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  -> 
( A  tX  C
)  C_  ( B  tX  D ) )

Proof of Theorem txss12
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . . . 5  |-  ran  (
x  e.  B , 
y  e.  D  |->  ( x  X.  y ) )  =  ran  (
x  e.  B , 
y  e.  D  |->  ( x  X.  y ) )
21txbasex 20233 . . . 4  |-  ( ( B  e.  V  /\  D  e.  W )  ->  ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) )  e. 
_V )
32adantr 463 . . 3  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  ->  ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y
) )  e.  _V )
4 resmpt2 6373 . . . . . 6  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) )  |`  ( A  X.  C
) )  =  ( x  e.  A , 
y  e.  C  |->  ( x  X.  y ) ) )
5 resss 5285 . . . . . 6  |-  ( ( x  e.  B , 
y  e.  D  |->  ( x  X.  y ) )  |`  ( A  X.  C ) )  C_  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y
) )
64, 5syl6eqssr 3540 . . . . 5  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( x  e.  A ,  y  e.  C  |->  ( x  X.  y
) )  C_  (
x  e.  B , 
y  e.  D  |->  ( x  X.  y ) ) )
76adantl 464 . . . 4  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  -> 
( x  e.  A ,  y  e.  C  |->  ( x  X.  y
) )  C_  (
x  e.  B , 
y  e.  D  |->  ( x  X.  y ) ) )
8 rnss 5220 . . . 4  |-  ( ( x  e.  A , 
y  e.  C  |->  ( x  X.  y ) )  C_  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) )  ->  ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y
) )  C_  ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y
) ) )
97, 8syl 16 . . 3  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  ->  ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y
) )  C_  ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y
) ) )
10 tgss 19637 . . 3  |-  ( ( ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) )  e. 
_V  /\  ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y ) ) 
C_  ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) ) )  ->  ( topGen `  ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y
) ) )  C_  ( topGen `  ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) ) ) )
113, 9, 10syl2anc 659 . 2  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  -> 
( topGen `  ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y ) ) )  C_  ( topGen ` 
ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) ) ) )
12 ssexg 4583 . . . . 5  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
13 ssexg 4583 . . . . 5  |-  ( ( C  C_  D  /\  D  e.  W )  ->  C  e.  _V )
14 eqid 2454 . . . . . 6  |-  ran  (
x  e.  A , 
y  e.  C  |->  ( x  X.  y ) )  =  ran  (
x  e.  A , 
y  e.  C  |->  ( x  X.  y ) )
1514txval 20231 . . . . 5  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  tX  C
)  =  ( topGen ` 
ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y ) ) ) )
1612, 13, 15syl2an 475 . . . 4  |-  ( ( ( A  C_  B  /\  B  e.  V
)  /\  ( C  C_  D  /\  D  e.  W ) )  -> 
( A  tX  C
)  =  ( topGen ` 
ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y ) ) ) )
1716an4s 824 . . 3  |-  ( ( ( A  C_  B  /\  C  C_  D )  /\  ( B  e.  V  /\  D  e.  W ) )  -> 
( A  tX  C
)  =  ( topGen ` 
ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y ) ) ) )
1817ancoms 451 . 2  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  -> 
( A  tX  C
)  =  ( topGen ` 
ran  ( x  e.  A ,  y  e.  C  |->  ( x  X.  y ) ) ) )
191txval 20231 . . 3  |-  ( ( B  e.  V  /\  D  e.  W )  ->  ( B  tX  D
)  =  ( topGen ` 
ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) ) ) )
2019adantr 463 . 2  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  -> 
( B  tX  D
)  =  ( topGen ` 
ran  ( x  e.  B ,  y  e.  D  |->  ( x  X.  y ) ) ) )
2111, 18, 203sstr4d 3532 1  |-  ( ( ( B  e.  V  /\  D  e.  W
)  /\  ( A  C_  B  /\  C  C_  D ) )  -> 
( A  tX  C
)  C_  ( B  tX  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461    X. cxp 4986   ran crn 4989    |` cres 4990   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   topGenctg 14927    tX ctx 20227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-topgen 14933  df-tx 20229
This theorem is referenced by: (None)
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