Theorem xpv 38766

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 5655	. 2 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
2		vex 3460	. . . 4 ⊢ 𝑦 ∈ V
3		iba 535	. . . 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 5169	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
6	1, 5	eqtr4i 2790	1 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456  {copab 5164   × cxp 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-opab 5165  df-xp 5655

This theorem is referenced by:  vxp  38767