Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  coi2 GIF version

Theorem coi2 4837
 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 4520 . . 3 (𝐴 ∘ I ) = ( I ∘ 𝐴)
2 relcnv 4703 . . . . 5 Rel 𝐴
3 coi1 4836 . . . . 5 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 7 . . . 4 (𝐴 ∘ I ) = 𝐴
54cnveqi 4510 . . 3 (𝐴 ∘ I ) = 𝐴
61, 5eqtr3i 2062 . 2 ( I ∘ 𝐴) = 𝐴
7 dfrel2 4771 . . 3 (Rel 𝐴𝐴 = 𝐴)
8 cnvi 4728 . . . 4 I = I
9 coeq2 4494 . . . . 5 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
10 coeq1 4493 . . . . 5 ( I = I → ( I ∘ 𝐴) = ( I ∘ 𝐴))
119, 10sylan9eq 2092 . . . 4 ((𝐴 = 𝐴 I = I ) → ( I ∘ 𝐴) = ( I ∘ 𝐴))
128, 11mpan2 401 . . 3 (𝐴 = 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
137, 12sylbi 114 . 2 (Rel 𝐴 → ( I ∘ 𝐴) = ( I ∘ 𝐴))
147biimpi 113 . 2 (Rel 𝐴𝐴 = 𝐴)
156, 13, 143eqtr3a 2096 1 (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   I cid 4025  ◡ccnv 4344   ∘ ccom 4349  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354 This theorem is referenced by:  relcoi2  4848  funi  4932  fcoi2  5071
 Copyright terms: Public domain W3C validator