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Theorem coi2 4934
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 4609 . . 3 (𝐴 ∘ I ) = ( I ∘ 𝐴)
2 relcnv 4797 . . . . 5 Rel 𝐴
3 coi1 4933 . . . . 5 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 7 . . . 4 (𝐴 ∘ I ) = 𝐴
54cnveqi 4599 . . 3 (𝐴 ∘ I ) = 𝐴
61, 5eqtr3i 2110 . 2 ( I ∘ 𝐴) = 𝐴
7 dfrel2 4868 . . 3 (Rel 𝐴𝐴 = 𝐴)
8 cnvi 4823 . . . 4 I = I
9 coeq2 4582 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 4581 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2140 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 416 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
137, 12sylbi 119 . 2 (Rel 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
147biimpi 118 . 2 (Rel 𝐴𝐴 = 𝐴)
156, 13, 143eqtr3a 2144 1 (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289   I cid 4106  ccnv 4427  ccom 4432  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437
This theorem is referenced by:  relcoi2  4948  funi  5032  fcoi2  5176
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