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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
StepHypRef 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
42, 3pm3.2i 474 . . . 4 (𝑥 ∈ V ∧ 𝑦 ∈ V)
54biantrur 534 . . 3 (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶))
65oprabbii 7229 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)}
71, 6eqtr4i 2764 1 (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}
Colors of variables: wff setvar class
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
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