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Theorem coi2 5813
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 5741 . 2 (Rel 𝐴𝐴 = 𝐴)
2 cnvco 5463 . . . 4 (𝐴 ∘ I ) = ( I ∘ 𝐴)
3 relcnv 5661 . . . . . 6 Rel 𝐴
4 coi1 5812 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
53, 4ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
65cnveqi 5452 . . . 4 (𝐴 ∘ I ) = 𝐴
72, 6eqtr3i 2784 . . 3 ( I ∘ 𝐴) = 𝐴
8 cnvi 5695 . . . 4 I = I
9 coeq2 5436 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 5435 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2814 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 709 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
13 id 22 . . 3 (𝐴 = 𝐴𝐴 = 𝐴)
147, 12, 133eqtr3a 2818 . 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 1632   I cid 5173  ccnv 5265  ccom 5270  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275
This theorem is referenced by:  relcoi2  5824  funi  6081  fcoi2  6240
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