Theorem xpv 38997

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 5665	. 2 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
2		vex 3457	. . . 4 ⊢ 𝑦 ∈ V
3		iba 537	. . . 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 5176	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
6	1, 5	eqtr4i 2788	1 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  {copab 5171   × cxp 5657

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 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-opab 5172  df-xp 5665

This theorem is used by:  vxp  38998