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Theorem mpo0v 7222
 Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) (Proof shortened by AV, 27-Jan-2024.)
Assertion
Ref Expression
mpo0v (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
Distinct variable groups:   𝑥,𝐵   𝑦,𝐵
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem mpo0v
StepHypRef Expression
1 eqid 2822 . . 3 ∅ = ∅
21orci 862 . 2 (∅ = ∅ ∨ 𝐵 = ∅)
3 0mpo0 7221 . 2 ((∅ = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅)
42, 3ax-mp 5 1 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 844   = wceq 1538  ∅c0 4265   ∈ cmpo 7142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-dif 3911  df-nul 4266  df-oprab 7144  df-mpo 7145 This theorem is referenced by: (None)
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