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Theorem cossxp 5378
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 5353 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5376 . . 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 5114 . . 3  |-  dom  ( A  o.  B )  C_ 
dom  B
5 rncoss 5115 . . 3  |-  ran  ( A  o.  B )  C_ 
ran  A
6 xpss12 4960 . . 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 676 . 2  |-  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  C_  ( dom  B  X.  ran  A )
83, 7sstri 3479 1  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3442    X. cxp 4852   dom cdm 4854   ran crn 4855    o. ccom 4858   Rel wrel 4859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865
This theorem is referenced by:  coexg  6758  tposssxp  6985  metustexhalf  21502
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