Theorem elco 38198

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 5667	. . 3 ⊢ (𝐶 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)}
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ (𝐶 ∘ 𝐵) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)})
3		elopab 5508	. 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 1570  ∃wex 1812   ∈ wcel 2147  ⟨cop 4840   class class class wbr 5108  {copab 5173   ∘ 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-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-op 4841  df-opab 5174  df-co 5667

This theorem is used by: (None)