Theorem mpov 7272

Description: Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

Assertion

Ref	Expression
mpov	⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝑧,𝐶

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem mpov

Step	Hyp	Ref	Expression
1		df-mpo 7169	. 2 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)}
2		vex 3401	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3401	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	pm3.2i 474	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
5	4	biantrur 534	. . 3 ⊢ (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶))
6	5	oprabbii 7229	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)}
7	1, 6	eqtr4i 2764	1 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}

Syntax hints:   ∧ wa 399   = wceq 1542   ∈ wcel 2113  Vcvv 3397  {coprab 7165   ∈ cmpo 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-12 2178  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-oprab 7168  df-mpo 7169

This theorem is referenced by:  1st2val  7735  2nd2val  7736