Theorem coi1 2665

Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.

Assertion

Ref	Expression
coi1	⊢ (Rel A → (A ∘ I) = A)

Proof of Theorem coi1

Step	Hyp	Ref	Expression
1		relco 2658	. 2 ⊢ Rel (A ∘ I)
2		visset 1350	. . . . . . 7 ⊢ x ∈ V
3		visset 1350	. . . . . . 7 ⊢ y ∈ V
4	2, 3	opelco 2509	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (A ∘ I) ↔ ∃z(xIz ∧ zAy))
5		visset 1350	. . . . . . . . . 10 ⊢ z ∈ V
6	2, 5	ideq 2127	. . . . . . . . 9 ⊢ (xIz ↔ x = z)
7		cleqcom 1103	. . . . . . . . 9 ⊢ (x = z ↔ z = x)
8	6, 7	bitr 151	. . . . . . . 8 ⊢ (xIz ↔ z = x)
9	8	anbi1i 368	. . . . . . 7 ⊢ ((xIz ∧ zAy) ↔ (z = x ∧ zAy))
10	9	biex 733	. . . . . 6 ⊢ (∃z(xIz ∧ zAy) ↔ ∃z(z = x ∧ zAy))
11		breq1 2065	. . . . . . 7 ⊢ (z = x → (zAy ↔ xAy))
12	2, 11	ceqsexv 1371	. . . . . 6 ⊢ (∃z(z = x ∧ zAy) ↔ xAy)
13	4, 10, 12	3bitr 155	. . . . 5 ⊢ (⟨x, y⟩ ∈ (A ∘ I) ↔ xAy)
14		df-br 2063	. . . . 5 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
15	13, 14	bitr 151	. . . 4 ⊢ (⟨x, y⟩ ∈ (A ∘ I) ↔ ⟨x, y⟩ ∈ A)
16	15	gen2 681	. . 3 ⊢ ∀x∀y(⟨x, y⟩ ∈ (A ∘ I) ↔ ⟨x, y⟩ ∈ A)
17		cleqrel 2483	. . 3 ⊢ ((Rel (A ∘ I) ∧ Rel A) → ((A ∘ I) = A ↔ ∀x∀y(⟨x, y⟩ ∈ (A ∘ I) ↔ ⟨x, y⟩ ∈ A)))
18	16, 17	mpbiri 169	. 2 ⊢ ((Rel (A ∘ I) ∧ Rel A) → (A ∘ I) = A)
19	1, 18	mpan 518	1 ⊢ (Rel A → (A ∘ I) = A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  Icid 2057   ∘ ccom 2414  Rel wrel 2415

This theorem is referenced by:  coi2 2666

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-co 2427

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