Theorem coi1 4836

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 4819	. 2 ⊢ Rel (𝐴 ∘ I )
2		vex 2560	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2560	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelco 4507	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦))
5		vex 2560	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
6	5	ideq 4488	. . . . . . . . 9 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
7		equcom 1593	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
8	6, 7	bitri 173	. . . . . . . 8 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥)
9	8	anbi1i 431	. . . . . . 7 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
10	9	exbii 1496	. . . . . 6 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
11		breq1 3767	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
12	2, 11	ceqsexv 2593	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
13	10, 12	bitri 173	. . . . 5 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
14	4, 13	bitri 173	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ 𝑥𝐴𝑦)
15		df-br 3765	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	14, 15	bitri 173	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ I ) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
17	16	eqrelriv 4433	. 2 ⊢ ((Rel (𝐴 ∘ I ) ∧ Rel 𝐴) → (𝐴 ∘ I ) = 𝐴)
18	1, 17	mpan 400	1 ⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764   I cid 4025   ∘ ccom 4349  Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-co 4354

This theorem is referenced by:  coi2  4837  coires1  4838  relcoi1  4849  fcoi1  5070