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Theorem xptrrel 12983
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 3620 . . . . . . . 8  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  dom  ( A  X.  B )
2 dmxpss 5225 . . . . . . . 8  |-  dom  ( A  X.  B )  C_  A
31, 2sstri 3411 . . . . . . 7  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  A
4 inss2 3621 . . . . . . . 8  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  ran  ( A  X.  B )
5 rnxpss 5226 . . . . . . . 8  |-  ran  ( A  X.  B )  C_  B
64, 5sstri 3411 . . . . . . 7  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  B
73, 6ssini 3623 . . . . . 6  |-  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  ( A  i^i  B )
8 eqimss 3454 . . . . . 6  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  B )  C_  (/) )
97, 8syl5ss 3413 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) ) 
C_  (/) )
10 ss0 3733 . . . . 5  |-  ( ( dom  ( A  X.  B )  i^i  ran  ( A  X.  B
) )  C_  (/)  ->  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) )  =  (/) )
119, 10syl 17 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( dom  ( A  X.  B
)  i^i  ran  ( A  X.  B ) )  =  (/) )
1211coemptyd 12982 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  B )  o.  ( A  X.  B ) )  =  (/) )
13 0ss 3731 . . 3  |-  (/)  C_  ( A  X.  B )
1412, 13syl6eqss 3452 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  B )  o.  ( A  X.  B ) )  C_  ( A  X.  B
) )
15 df-ne 2596 . . . . 5  |-  ( ( A  i^i  B )  =/=  (/)  <->  -.  ( A  i^i  B )  =  (/) )
1615biimpri 209 . . . 4  |-  ( -.  ( A  i^i  B
)  =  (/)  ->  ( A  i^i  B )  =/=  (/) )
1716xpcoidgend 12978 . . 3  |-  ( -.  ( A  i^i  B
)  =  (/)  ->  (
( A  X.  B
)  o.  ( A  X.  B ) )  =  ( A  X.  B ) )
18 ssid 3421 . . 3  |-  ( A  X.  B )  C_  ( A  X.  B
)
1917, 18syl6eqss 3452 . 2  |-  ( -.  ( A  i^i  B
)  =  (/)  ->  (
( A  X.  B
)  o.  ( A  X.  B ) ) 
C_  ( A  X.  B ) )
2014, 19pm2.61i 167 1  |-  ( ( A  X.  B )  o.  ( A  X.  B ) )  C_  ( A  X.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    =/= wne 2594    i^i cin 3373    C_ wss 3374   (/)c0 3699    X. cxp 4789   dom cdm 4791   ran crn 4792    o. ccom 4795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pr 4598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-br 4362  df-opab 4421  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803
This theorem is referenced by:  trclublem  12998  trclubgNEW  36138  trclexi  36140  cnvtrcl0  36146  xpintrreld  36171  trrelsuperreldg  36173  trrelsuperrel2dg  36176
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