Theorem xpv 38861

Description: Cartesian product of a class and the universe. (Contributed by Peter Mazsa, 6-Oct-2020.)

Assertion

Ref	Expression
xpv	⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem xpv

Step	Hyp	Ref	Expression
1		df-xp 5671	. 2 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
2		vex 3466	. . . 4 ⊢ 𝑦 ∈ V
3		iba 536	. . . 4 ⊢ (𝑦 ∈ V → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V))
5	4	opabbii 5183	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
6	1, 5	eqtr4i 2796	1 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  Vcvv 3462  {copab 5178   × cxp 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-opab 5179  df-xp 5671

This theorem is referenced by:  vxp  38862