Theorem coi2 4857

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

Step	Hyp	Ref	Expression
1		cnvco 4538	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
2		relcnv 4723	. . . . 5 ⊢ Rel ◡𝐴
3		coi1 4856	. . . . 5 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
4	2, 3	ax-mp 7	. . . 4 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
5	4	cnveqi 4528	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
6	1, 5	eqtr3i 2103	. 2 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
7		dfrel2 4791	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8		cnvi 4748	. . . 4 ⊢ ◡ I = I
9		coeq2 4512	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 4511	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2133	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 415	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13	7, 12	sylbi 119	. 2 ⊢ (Rel 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
14	7	biimpi 118	. 2 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴)
15	6, 13, 14	3eqtr3a 2137	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1284   I cid 4043  ^◡ccnv 4362   ∘ ccom 4367  Rel wrel 4368

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372

This theorem is referenced by:  relcoi2  4868  funi  4952  fcoi2  5091