Theorem coi1res 38066

Description: Precomposing with the identity is the same as taking the relation part of a class. (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 3454	. . . . . . . 8 ⊢ 𝑧 ∈ V
2	1	ideq 5830	. . . . . . 7 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
3		equcom 2050	. . . . . . 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 1880	. . . 4 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦))
7		breq1 5105	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
8	7	equsexvw 2037	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
9	6, 8	bitri 278	. . 3 ⊢ (∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑥𝐴𝑦)
10	9	opabbii 5171	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
11		df-co 5659	. 2 ⊢ (𝐴 ∘ I ) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥 I 𝑧 ∧ 𝑧𝐴𝑦)}
12		relpart 38058	. 2 ⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
13	10, 11, 12	3eqtr4i 2793	1 ⊢ (𝐴 ∘ I ) = (𝐴 ↾ V)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∃wex 1811  Vcvv 3450   class class class wbr 5102  {copab 5166   I cid 5544   ↾ cres 5652   ∘ ccom 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-co 5659  df-res 5662

This theorem is used by:  bj-coi1  38067