Theorem xpv 38432

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 5629	. 2 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
2		vex 3443	. . . 4 ⊢ 𝑦 ∈ V
3		iba 527	. . . 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 5164	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
6	1, 5	eqtr4i 2761	1 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3439  {copab 5159   × cxp 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3441  df-opab 5160  df-xp 5629

This theorem is referenced by:  vxp  38433