Theorem mpo0v 7425

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 2737	. . 3 ⊢ ∅ = ∅
2	1	orci 863	. 2 ⊢ (∅ = ∅ ∨ 𝐵 = ∅)
3		0mpo0 7424	. 2 ⊢ ((∅ = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)
4	2, 3	ax-mp 5	1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Syntax hints:   ∨ wo 845   = wceq 1541  ∅c0 4273   ∈ cmpo 7343

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-dif 3904  df-nul 4274  df-oprab 7345  df-mpo 7346

This theorem is referenced by: (None)