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Theorem cossxp 4873
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 4849 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 4871 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( A  o.  B )  C_  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
) )
31, 2ax-mp 7 . 2  |-  ( A  o.  B )  C_  ( dom  ( A  o.  B )  X.  ran  ( A  o.  B
) )
4 dmcoss 4629 . . 3  |-  dom  ( A  o.  B )  C_ 
dom  B
5 rncoss 4630 . . 3  |-  ran  ( A  o.  B )  C_ 
ran  A
6 xpss12 4473 . . 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 417 . 2  |-  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  C_  ( dom  B  X.  ran  A )
83, 7sstri 3009 1  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
Colors of variables: wff set class
Syntax hints:    C_ wss 2974    X. cxp 4369   dom cdm 4371   ran crn 4372    o. ccom 4375   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382
This theorem is referenced by:  coexg  4892  tposssxp  5898
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