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Theorem coi1 5528
 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 5510 . 2
2 vex 3112 . . . . . 6
3 vex 3112 . . . . . 6
42, 3opelco 5179 . . . . 5
5 vex 3112 . . . . . . . . . 10
65ideq 5160 . . . . . . . . 9
7 equcom 1794 . . . . . . . . 9
86, 7bitri 249 . . . . . . . 8
98anbi1i 695 . . . . . . 7
109exbii 1667 . . . . . 6
11 breq1 4455 . . . . . . 7
122, 11ceqsexv 3146 . . . . . 6
1310, 12bitri 249 . . . . 5
144, 13bitri 249 . . . 4
15 df-br 4453 . . . 4
1614, 15bitri 249 . . 3
1716eqrelriv 5101 . 2
181, 17mpan 670 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035   class class class wbr 4452   cid 4795  o.ccom 5008  Relwrel 5009 This theorem is referenced by:  coi2  5529  coires1  5530  relcoi1  5541  fcoi1  5764  mvdco  16470  cocnv  30216 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-co 5013
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