Theorem coi1 6282

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
coi1	⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Proof of Theorem coi1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6126	. 2 ⊢ Rel (𝐴 ∘ I )
2		vex 3484	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3484	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelco 5882	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦))
5		vex 3484	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
6	5	ideq 5863	. . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
7		equcom 2017	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
8	6, 7	bitri 275	. . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥)
9	8	anbi1i 624	. . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
10	9	exbii 1848	. . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
11		breq1 5146	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
12	11	equsexvw 2004	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
13	10, 12	bitri 275	. . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
14	4, 13	bitri 275	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15		df-br 5144	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	14, 15	bitri 275	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
17	16	eqrelriv 5799	. 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
18	1, 17	mpan 690	1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ⟨cop 4632   class class class wbr 5143   I cid 5577   ∘ ccom 5689  Rel wrel 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-co 5694

This theorem is referenced by:  coi2  6283  coires1  6284  fcoi1  6782  mvdco  19463  cocnv  37732