Theorem coi1 5119

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 5102	. 2 ⊢ Rel (𝐴 ∘ I )
2		vex 2729	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2729	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelco 4776	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦))
5		vex 2729	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
6	5	ideq 4756	. . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
7		equcom 1694	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
8	6, 7	bitri 183	. . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥)
9	8	anbi1i 454	. . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
10	9	exbii 1593	. . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
11		breq1 3985	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
12	2, 11	ceqsexv 2765	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
13	10, 12	bitri 183	. . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
14	4, 13	bitri 183	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15		df-br 3983	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	14, 15	bitri 183	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
17	16	eqrelriv 4697	. 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
18	1, 17	mpan 421	1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ⟨cop 3579   class class class wbr 3982   I cid 4266   ∘ ccom 4608  Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-co 4613

This theorem is referenced by:  coi2  5120  coires1  5121  relcoi1  5135  fcoi1  5368