Theorem mpo0v 7452

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 866	. 2 ⊢ (∅ = ∅ ∨ 𝐵 = ∅)
3		0mpo0 7451	. 2 ⊢ ((∅ = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)
4	2, 3	ax-mp 5	1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Syntax hints:   ∨ wo 848   = wceq 1542  ∅c0 4287   ∈ cmpo 7370

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-dif 3906  df-nul 4288  df-oprab 7372  df-mpo 7373

This theorem is referenced by: (None)