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Theorem cossxp 5384
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 5360 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5382 . . 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 5127 . . 3  |-  dom  ( A  o.  B )  C_ 
dom  B
5 rncoss 5128 . . 3  |-  ran  ( A  o.  B )  C_ 
ran  A
6 xpss12 4973 . . 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 3349 1  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
Colors of variables: wff set class
Syntax hints:    C_ wss 3312    X. cxp 4868   dom cdm 4870   ran crn 4871    o. ccom 4874   Rel wrel 4875
This theorem is referenced by:  coexg  5404  tposssxp  6475  metustexhalfOLD  18585  metustexhalf  18586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881
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