Theorem coi2res 38193

Description: Postcomposing with the identity is the same as taking the relation part of a class. See also coi1res 38192 and coires2 6287. (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
coi2res	⊢ ( I ∘ 𝐴) = (𝐴 ↾ V)

Proof of Theorem coi2res

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

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . . . 8 ⊢ 𝑦 ∈ V
2	1	ideq 5838	. . . . . . 7 ⊢ (𝑧 I 𝑦 ↔ 𝑧 = 𝑦)
3	2	anbi2i 635	. . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧 I 𝑦) ↔ (𝑥𝐴𝑧 ∧ 𝑧 = 𝑦))
4		ancom 466	. . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑥𝐴𝑧))
5	3, 4	bitri 278	. . . . 5 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧 I 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑥𝐴𝑧))
6	5	exbii 1881	. . . 4 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧 I 𝑦) ↔ ∃𝑧(𝑧 = 𝑦 ∧ 𝑥𝐴𝑧))
7		breq2 5110	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥𝐴𝑧 ↔ 𝑥𝐴𝑦))
8	7	equsexvw 2040	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑦 ∧ 𝑥𝐴𝑧) ↔ 𝑥𝐴𝑦)
9	6, 8	bitri 278	. . 3 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧 I 𝑦) ↔ 𝑥𝐴𝑦)
10	9	opabbii 5176	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧 I 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
11		df-co 5665	. 2 ⊢ ( I ∘ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧 I 𝑦)}
12		relpart 38179	. 2 ⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
13	10, 11, 12	3eqtr4i 2796	1 ⊢ ( I ∘ 𝐴) = (𝐴 ↾ V)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812  Vcvv 3454   class class class wbr 5106  {copab 5171   I cid 5550   ↾ cres 5658   ∘ ccom 5660

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 5253  ax-pr 5396

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 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-co 5665  df-res 5668

This theorem is used by:  coi2ss  38195