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Theorem coi2 5690
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 5618 . 2 (Rel 𝐴𝐴 = 𝐴)
2 cnvco 5340 . . . 4 (𝐴 ∘ I ) = ( I ∘ 𝐴)
3 relcnv 5538 . . . . . 6 Rel 𝐴
4 coi1 5689 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
53, 4ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
65cnveqi 5329 . . . 4 (𝐴 ∘ I ) = 𝐴
72, 6eqtr3i 2675 . . 3 ( I ∘ 𝐴) = 𝐴
8 cnvi 5572 . . . 4 I = I
9 coeq2 5313 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 5312 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2705 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 707 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
13 id 22 . . 3 (𝐴 = 𝐴𝐴 = 𝐴)
147, 12, 133eqtr3a 2709 . 2 (𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴)
151, 14sylbi 207 1 (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523   I cid 5052  ccnv 5142  ccom 5147  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152
This theorem is referenced by:  relcoi2  5701  funi  5958  fcoi2  6117
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