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Theorem coi2 6253
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 6179 . 2 (Rel 𝐴𝐴 = 𝐴)
2 cnvco 5876 . . . 4 (𝐴 ∘ I ) = ( I ∘ 𝐴)
3 relcnv 6094 . . . . . 6 Rel 𝐴
4 coi1 6252 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
53, 4ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
65cnveqi 5865 . . . 4 (𝐴 ∘ I ) = 𝐴
72, 6eqtr3i 2754 . . 3 ( I ∘ 𝐴) = 𝐴
8 cnvi 6132 . . . 4 I = I
9 coeq2 5849 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 5848 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2784 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 688 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
13 id 22 . . 3 (𝐴 = 𝐴𝐴 = 𝐴)
147, 12, 133eqtr3a 2788 . 2 (𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴)
151, 14sylbi 216 1 (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   I cid 5564  ccnv 5666  ccom 5671  Rel wrel 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676
This theorem is referenced by:  relcoi2  6267  funi  6571  fcoi2  6757
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