Theorem coi1 4889

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 4872	. 2 ⊢ Rel (𝐴 ∘ I )
2		vex 2614	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2614	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelco 4558	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦))
5		vex 2614	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
6	5	ideq 4539	. . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
7		equcom 1635	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
8	6, 7	bitri 182	. . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥)
9	8	anbi1i 446	. . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
10	9	exbii 1537	. . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
11		breq1 3809	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
12	2, 11	ceqsexv 2648	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
13	10, 12	bitri 182	. . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
14	4, 13	bitri 182	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15		df-br 3807	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	14, 15	bitri 182	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
17	16	eqrelriv 4482	. 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
18	1, 17	mpan 415	1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285  ∃wex 1422   ∈ wcel 1434  ⟨cop 3420   class class class wbr 3806   I cid 4073   ∘ ccom 4398  Rel wrel 4399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3994

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-id 4078  df-xp 4400  df-rel 4401  df-co 4403

This theorem is referenced by:  coi2  4890  coires1  4891  relcoi1  4902  fcoi1  5124