Theorem elco 37940

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 5660	. . 3 ⊢ (𝐶 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)}
2	1	eleq2i 2853	. 2 ⊢ (𝐴 ∈ (𝐶 ∘ 𝐵) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)})
3		elopab 5501	. 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 2145  ⟨cop 4590   class class class wbr 5103  {copab 5167   ∘ ccom 5655

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-un 3904  df-in 3906  df-ss 3916  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-co 5660

This theorem is used by:  coi1in  37941