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Theorem coi1 5339
 Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
coi1

Proof of Theorem coi1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5322 . 2
2 vex 2916 . . . . . 6
3 vex 2916 . . . . . 6
42, 3opelco 4998 . . . . 5
5 vex 2916 . . . . . . . . . 10
65ideq 4979 . . . . . . . . 9
7 equcom 1688 . . . . . . . . 9
86, 7bitri 241 . . . . . . . 8
98anbi1i 677 . . . . . . 7
109exbii 1589 . . . . . 6
11 breq1 4170 . . . . . . 7
122, 11ceqsexv 2948 . . . . . 6
1310, 12bitri 241 . . . . 5
144, 13bitri 241 . . . 4
15 df-br 4168 . . . 4
1614, 15bitri 241 . . 3
1716eqrelriv 4923 . 2
181, 17mpan 652 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1547   wceq 1649   wcel 1721  cop 3774   class class class wbr 4167   cid 4448   ccom 4836   wrel 4837 This theorem is referenced by:  coi2  5340  coires1  5341  relcoi1  5352  fcoi1  5571  cocnv  26134  mvdco  27073 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2382  ax-sep 4285  ax-nul 4293  ax-pr 4358 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2256  df-mo 2257  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2526  df-ne 2566  df-ral 2668  df-rex 2669  df-rab 2672  df-v 2915  df-dif 3280  df-un 3282  df-in 3284  df-ss 3291  df-nul 3586  df-if 3697  df-sn 3777  df-pr 3778  df-op 3780  df-br 4168  df-opab 4222  df-id 4453  df-xp 4838  df-rel 4839  df-co 4841
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