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Theorem mpov 7523
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
StepHypRef Expression
1 df-mpo 7417 . 2 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)}
2 vex 3477 . . . . 5 𝑥 ∈ V
3 vex 3477 . . . . 5 𝑦 ∈ V
42, 3pm3.2i 470 . . . 4 (𝑥 ∈ V ∧ 𝑦 ∈ V)
54biantrur 530 . . 3 (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶))
65oprabbii 7479 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)}
71, 6eqtr4i 2762 1 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  {coprab 7413  cmpo 7414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  1st2val  8007  2nd2val  8008
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