Theorem mpo0v 7441

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 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Distinct variable groups:   𝑥,𝐵   𝑦,𝐵

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem mpo0v

Step	Hyp	Ref	Expression
1		eqid 2739	. . 3 ⊢ ∅ = ∅
2	1	orci 871	. 2 ⊢ (∅ = ∅ ∨ 𝐵 = ∅)
3		0mpo0 7440	. 2 ⊢ ((∅ = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)
4	2, 3	ax-mp 5	1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Syntax hints:   ∨ wo 853   = wceq 1547  ∅c0 4262   ∈ cmpo 7359

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 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-dif 3886  df-nul 4263  df-oprab 7361  df-mpo 7362

This theorem is referenced by: (None)