Theorem elco 38053

Description: Membership in a composition. (Contributed by BJ, 16-Aug-2026.)

Assertion

Ref	Expression
elco	⊢ (𝐴 ∈ (𝐶 ∘ 𝐵) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem elco

Step	Hyp	Ref	Expression
1		df-co 5659	. . 3 ⊢ (𝐶 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)}
2	1	eleq2i 2852	. 2 ⊢ (𝐴 ∈ (𝐶 ∘ 𝐵) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)})
3		elopab 5500	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)))
4	2, 3	bitri 278	1 ⊢ (𝐴 ∈ (𝐶 ∘ 𝐵) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ⟨cop 4834   class class class wbr 5102  {copab 5166   ∘ 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-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-un 4151  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-co 5659

This theorem is used by:  coi1in  38054