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Theorem coi2 4865
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 cnvco 4548 . . 3 (𝐴 ∘ I ) = ( I ∘ 𝐴)
2 relcnv 4731 . . . . 5 Rel 𝐴
3 coi1 4864 . . . . 5 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 7 . . . 4 (𝐴 ∘ I ) = 𝐴
54cnveqi 4538 . . 3 (𝐴 ∘ I ) = 𝐴
61, 5eqtr3i 2078 . 2 ( I ∘ 𝐴) = 𝐴
7 dfrel2 4799 . . 3 (Rel 𝐴𝐴 = 𝐴)
8 cnvi 4756 . . . 4 I = I
9 coeq2 4522 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 4521 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2108 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 409 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
137, 12sylbi 118 . 2 (Rel 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
147biimpi 117 . 2 (Rel 𝐴𝐴 = 𝐴)
156, 13, 143eqtr3a 2112 1 (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259   I cid 4053  ccnv 4372  ccom 4377  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382
This theorem is referenced by:  relcoi2  4876  funi  4960  fcoi2  5099
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