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Theorem coi2 6216
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 6142 . 2 (Rel 𝐴𝐴 = 𝐴)
2 cnvco 5842 . . . 4 (𝐴 ∘ I ) = ( I ∘ 𝐴)
3 relcnv 6057 . . . . . 6 Rel 𝐴
4 coi1 6215 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
53, 4ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
65cnveqi 5831 . . . 4 (𝐴 ∘ I ) = 𝐴
72, 6eqtr3i 2763 . . 3 ( I ∘ 𝐴) = 𝐴
8 cnvi 6095 . . . 4 I = I
9 coeq2 5815 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 5814 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2793 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 690 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
13 id 22 . . 3 (𝐴 = 𝐴𝐴 = 𝐴)
147, 12, 133eqtr3a 2797 . 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 1542   I cid 5531  ccnv 5633  ccom 5638  Rel wrel 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643
This theorem is referenced by:  relcoi2  6230  funi  6534  fcoi2  6718
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