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Theorem cossxp 5333
Description: Composition as a subset of the cross 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 5309 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5331 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( A  o.  B )  C_  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
) )
31, 2ax-mp 8 . 2  |-  ( A  o.  B )  C_  ( dom  ( A  o.  B )  X.  ran  ( A  o.  B
) )
4 dmcoss 5076 . . 3  |-  dom  ( A  o.  B )  C_ 
dom  B
5 rncoss 5077 . . 3  |-  ran  ( A  o.  B )  C_ 
ran  A
6 xpss12 4922 . . 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 654 . 2  |-  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  C_  ( dom  B  X.  ran  A )
83, 7sstri 3301 1  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
Colors of variables: wff set class
Syntax hints:    C_ wss 3264    X. cxp 4817   dom cdm 4819   ran crn 4820    o. ccom 4823   Rel wrel 4824
This theorem is referenced by:  coexg  5353  tposssxp  6420  metustexhalf  18477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830
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