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Theorem cossxp 5536
Description: Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
cossxp  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )

Proof of Theorem cossxp
StepHypRef Expression
1 relco 5511 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5534 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( A  o.  B )  C_  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
) )
31, 2ax-mp 5 . 2  |-  ( A  o.  B )  C_  ( dom  ( A  o.  B )  X.  ran  ( A  o.  B
) )
4 dmcoss 5268 . . 3  |-  dom  ( A  o.  B )  C_ 
dom  B
5 rncoss 5269 . . 3  |-  ran  ( A  o.  B )  C_ 
ran  A
6 xpss12 5114 . . 3  |-  ( ( dom  ( A  o.  B )  C_  dom  B  /\  ran  ( A  o.  B )  C_  ran  A )  ->  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  C_  ( dom  B  X.  ran  A ) )
74, 5, 6mp2an 672 . 2  |-  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  C_  ( dom  B  X.  ran  A )
83, 7sstri 3518 1  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3481    X. cxp 5003   dom cdm 5005   ran crn 5006    o. ccom 5009   Rel wrel 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016
This theorem is referenced by:  coexg  6746  tposssxp  6971  metustexhalfOLD  20934  metustexhalf  20935
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