Theorem coi1 5259

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 5242	. 2 ⊢ Rel (𝐴 ∘ I )
2		vex 2806	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2806	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelco 4908	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦))
5		vex 2806	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
6	5	ideq 4888	. . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
7		equcom 1754	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
8	6, 7	bitri 184	. . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥)
9	8	anbi1i 458	. . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
10	9	exbii 1654	. . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
11		breq1 4096	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
12	2, 11	ceqsexv 2843	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
13	10, 12	bitri 184	. . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
14	4, 13	bitri 184	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15		df-br 4094	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	14, 15	bitri 184	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
17	16	eqrelriv 4825	. 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
18	1, 17	mpan 424	1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ⟨cop 3676   class class class wbr 4093   I cid 4391   ∘ 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-co 4740

This theorem is referenced by:  coi2  5260  coires1  5261  relcoi1  5275  fcoi1  5525