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Theorem xptrrel 12901
Description: The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)
Assertion
Ref Expression
xptrrel  |-  ( ( A  X.  B )  o.  ( A  X.  B ) )  C_  ( A  X.  B
)

Proof of Theorem xptrrel
StepHypRef Expression
1 inss1 3704 . . . . . . . 8  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  dom  ( A  X.  B )
2 dmxpss 5423 . . . . . . . 8  |-  dom  ( A  X.  B )  C_  A
31, 2sstri 3498 . . . . . . 7  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  A
4 inss2 3705 . . . . . . . 8  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  ran  ( A  X.  B )
5 rnxpss 5424 . . . . . . . 8  |-  ran  ( A  X.  B )  C_  B
64, 5sstri 3498 . . . . . . 7  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  B
73, 6ssini 3707 . . . . . 6  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  ( A  i^i  B )
8 eqimss 3541 . . . . . 6  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  B )  C_  (/) )
97, 8syl5ss 3500 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  (/) )
10 ss0 3815 . . . . 5  |-  ( ( dom  ( A  X.  B )  i^i  ran  ( A  X.  B
) )  C_  (/)  ->  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) )  =  (/) )
119, 10syl 16 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) )  =  (/) )
1211coemptyd 12900 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  B )  o.  ( A  X.  B ) )  =  (/) )
13 0ss 3813 . . 3  |-  (/)  C_  ( A  X.  B )
1412, 13syl6eqss 3539 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  B )  o.  ( A  X.  B ) )  C_  ( A  X.  B
) )
15 df-ne 2651 . . . . 5  |-  ( ( A  i^i  B )  =/=  (/)  <->  -.  ( A  i^i  B )  =  (/) )
1615biimpri 206 . . . 4  |-  ( -.  ( A  i^i  B
)  =  (/)  ->  ( A  i^i  B )  =/=  (/) )
1716xpcoidgend 12896 . . 3  |-  ( -.  ( A  i^i  B
)  =  (/)  ->  (
( A  X.  B
)  o.  ( A  X.  B ) )  =  ( A  X.  B ) )
18 ssid 3508 . . 3  |-  ( A  X.  B )  C_  ( A  X.  B
)
1917, 18syl6eqss 3539 . 2  |-  ( -.  ( A  i^i  B
)  =  (/)  ->  (
( A  X.  B
)  o.  ( A  X.  B ) ) 
C_  ( A  X.  B ) )
2014, 19pm2.61i 164 1  |-  ( ( A  X.  B )  o.  ( A  X.  B ) )  C_  ( A  X.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    =/= wne 2649    i^i cin 3460    C_ wss 3461   (/)c0 3783    X. cxp 4986   dom cdm 4988   ran crn 4989    o. ccom 4992
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
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-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000
This theorem is referenced by:  trclublem  12916  xpintrreld  38208  trrelsuperreldg  38211
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