Theorem mpov 7501

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 7392	. 2 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)}
2		vex 3451	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3451	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	pm3.2i 470	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
5	4	biantrur 530	. . 3 ⊢ (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶))
6	5	oprabbii 7456	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)}
7	1, 6	eqtr4i 2755	1 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  {coprab 7388   ∈ cmpo 7389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-oprab 7391  df-mpo 7392

This theorem is referenced by:  1st2val  7996  2nd2val  7997