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Theorem cossxp2 5074
 Description: The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
cossxp2.r
cossxp2.s
Assertion
Ref Expression
cossxp2

Proof of Theorem cossxp2
StepHypRef Expression
1 cossxp 5073 . 2
2 cossxp2.r . . . 4
3 dmxpss2 4983 . . . 4
42, 3syl 14 . . 3
5 cossxp2.s . . . 4
6 rnxpss2 4984 . . . 4
75, 6syl 14 . . 3
8 xpss12 4658 . . 3
94, 7, 8syl2anc 409 . 2
101, 9sstrid 3115 1
 Colors of variables: wff set class Syntax hints:   wi 4   wss 3078   cxp 4549   cdm 4551   crn 4552   ccom 4555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562 This theorem is referenced by: (None)
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