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Theorem cossxp 4786
Description: Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
cossxp  o.  C_  dom  X.  ran

Proof of Theorem cossxp
StepHypRef Expression
1 relco 4762 . . 3  Rel  o.
2 relssdmrn 4784 . . 3  Rel  o.  o.  C_  dom  o.  X.  ran  o.
31, 2ax-mp 7 . 2  o.  C_  dom  o.  X.  ran  o.
4 dmcoss 4544 . . 3  dom  o.  C_  dom
5 rncoss 4545 . . 3  ran  o.  C_  ran
6 xpss12 4388 . . 3  dom  o.  C_  dom  ran  o.  C_  ran  dom  o.  X.  ran  o.  C_  dom  X.  ran
74, 5, 6mp2an 402 . 2  dom  o.  X.  ran  o. 
C_  dom  X.  ran
83, 7sstri 2948 1  o.  C_  dom  X.  ran
Colors of variables: wff set class
Syntax hints:    C_ wss 2911    X. cxp 4286   dom cdm 4288   ran crn 4289    o. ccom 4292   Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299
This theorem is referenced by:  coexg  4805  tposssxp  5805
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