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Theorem coi2 5529
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
coi2

Proof of Theorem coi2
StepHypRef Expression
1 cnvco 5193 . . 3
2 relcnv 5379 . . . . 5
3 coi1 5528 . . . . 5
42, 3ax-mp 5 . . . 4
54cnveqi 5182 . . 3
61, 5eqtr3i 2488 . 2
7 dfrel2 5462 . . 3
8 cnvi 5415 . . . 4
9 coeq2 5166 . . . . 5
10 coeq1 5165 . . . . 5
119, 10sylan9eq 2518 . . . 4
128, 11mpan2 671 . . 3
137, 12sylbi 195 . 2
147biimpi 194 . 2
156, 13, 143eqtr3a 2522 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395   cid 4795  `'ccnv 5003  o.ccom 5008  Relwrel 5009
This theorem is referenced by:  relcoi2  5540  funi  5623  fcoi2  5765
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-cnv 5012  df-co 5013
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