Theorem coi2 5260

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 4921	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
2		relcnv 5121	. . . . 5 ⊢ Rel ◡𝐴
3		coi1 5259	. . . . 5 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
4	2, 3	ax-mp 5	. . . 4 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
5	4	cnveqi 4911	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
6	1, 5	eqtr3i 2254	. 2 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
7		dfrel2 5194	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8		cnvi 5148	. . . 4 ⊢ ◡ I = I
9		coeq2 4894	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 4893	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2284	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 425	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13	7, 12	sylbi 121	. 2 ⊢ (Rel 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
14	7	biimpi 120	. 2 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴)
15	6, 13, 14	3eqtr3a 2288	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1398   I cid 4391  ^◡ccnv 4730   ∘ ccom 4735  Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740

This theorem is referenced by:  relcoi2  5274  funi  5365  fcoi2  5526