Theorem brcoi 5852

Description: Ordered pair membership in a composition. (Contributed by BJ, 25-Oct-2026.)

Hypotheses

Ref	Expression
brcoi.1	⊢ 𝑋 ∈ V
brcoi.2	⊢ 𝑌 ∈ V
brcoi.3	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
brcoi	⊢ ((𝑋𝑅𝑌 ∧ 𝑌𝑆𝑍) → 𝑋(𝑆 ∘ 𝑅)𝑍)

Proof of Theorem brcoi

Step	Hyp	Ref	Expression
1		brcoi.1	. . 3 ⊢ 𝑋 ∈ V
2		brcoi.3	. . 3 ⊢ 𝑍 ∈ V
3		brcoi.2	. . 3 ⊢ 𝑌 ∈ V
4	1, 2, 3	3pm3.2i 1358	. 2 ⊢ (𝑋 ∈ V ∧ 𝑍 ∈ V ∧ 𝑌 ∈ V)
5		brcogw 5851	. 2 ⊢ (((𝑋 ∈ V ∧ 𝑍 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑋𝑅𝑌 ∧ 𝑌𝑆𝑍)) → 𝑋(𝑆 ∘ 𝑅)𝑍)
6	4, 5	mpan 703	1 ⊢ ((𝑋𝑅𝑌 ∧ 𝑌𝑆𝑍) → 𝑋(𝑆 ∘ 𝑅)𝑍)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106   ∘ ccom 5659

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 5252  ax-pr 5395

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-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 5171  df-co 5664

This theorem is used by:  idresdmsscocnv1  6274  coass  6281  ttrclss  9745  utop2nei  24596  utop3cls  24597  iunrelexpuztr  44909