Theorem coi2 6216

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		dfrel2 6141	. 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		cnvco 5829	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
3		relcnv 6057	. . . . . 6 ⊢ Rel ◡𝐴
4		coi1 6215	. . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
6	5	cnveqi 5818	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
7	2, 6	eqtr3i 2758	. . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
8		cnvi 6093	. . . 4 ⊢ ◡ I = I
9		coeq2 5802	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 5801	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2788	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 691	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13		id 22	. . 3 ⊢ (◡◡𝐴 = 𝐴 → ◡◡𝐴 = 𝐴)
14	7, 12, 13	3eqtr3a 2792	. 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴)
15	1, 14	sylbi 217	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   I cid 5513  ^◡ccnv 5618   ∘ ccom 5623  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628

This theorem is referenced by:  relcoi2  6229  funi  6518  fcoi2  6703