Theorem coi1res 38199

Description: Precomposing with the identity is the same as taking the relation part of a class. See also coires1 6288. (Contributed by BJ, 16-Aug-2026.)

Assertion

Ref	Expression
coi1res	⊢ (𝐴 ∘ I ) = (𝐴 ↾ V)

Proof of Theorem coi1res

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . . . 8 ⊢ 𝑧 ∈ V
2	1	ideq 5840	. . . . . . 7 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
3		equcom 2053	. . . . . . 7 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
4	2, 3	bitri 278	. . . . . 6 ⊢ (𝑥 I 𝑧 ↔ 𝑧 = 𝑥)
5	4	anbi1i 636	. . . . 5 ⊢ ((𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
6	5	exbii 1881	. . . 4 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
7		breq1 5111	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
8	7	equsexvw 2040	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
9	6, 8	bitri 278	. . 3 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
10	9	opabbii 5178	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
11		df-co 5667	. 2 ⊢ (𝐴 ∘ I ) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)}
12		relpart 38186	. 2 ⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
13	10, 11, 12	3eqtr4i 2796	1 ⊢ (𝐴 ∘ I ) = (𝐴 ↾ V)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812  Vcvv 3454   class class class wbr 5108  {copab 5173   I cid 5552   ↾ cres 5660   ∘ ccom 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-co 5667  df-res 5670

This theorem is used by:  coi1ss  38201  bj-coi1  38203