Theorem coi2 5127

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 4796	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
2		relcnv 4989	. . . . 5 ⊢ Rel ◡𝐴
3		coi1 5126	. . . . 5 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
4	2, 3	ax-mp 5	. . . 4 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
5	4	cnveqi 4786	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
6	1, 5	eqtr3i 2193	. 2 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
7		dfrel2 5061	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8		cnvi 5015	. . . 4 ⊢ ◡ I = I
9		coeq2 4769	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 4768	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2223	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 423	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13	7, 12	sylbi 120	. 2 ⊢ (Rel 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
14	7	biimpi 119	. 2 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴)
15	6, 13, 14	3eqtr3a 2227	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1348   I cid 4273  ^◡ccnv 4610   ∘ ccom 4615  Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620

This theorem is referenced by:  relcoi2  5141  funi  5230  fcoi2  5379