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Theorem coi2 6282
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 (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Proof of Theorem coi2
StepHypRef Expression
1 dfrel2 6208 . 2 (Rel 𝐴𝐴 = 𝐴)
2 cnvco 5895 . . . 4 (𝐴 ∘ I ) = ( I ∘ 𝐴)
3 relcnv 6121 . . . . . 6 Rel 𝐴
4 coi1 6281 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
53, 4ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
65cnveqi 5884 . . . 4 (𝐴 ∘ I ) = 𝐴
72, 6eqtr3i 2766 . . 3 ( I ∘ 𝐴) = 𝐴
8 cnvi 6160 . . . 4 I = I
9 coeq2 5868 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 5867 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2796 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 691 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
13 id 22 . . 3 (𝐴 = 𝐴𝐴 = 𝐴)
147, 12, 133eqtr3a 2800 . 2 (𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴)
151, 14sylbi 217 1 (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   I cid 5576  ccnv 5683  ccom 5688  Rel wrel 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693
This theorem is referenced by:  relcoi2  6296  funi  6597  fcoi2  6782
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