Theorem mpo0v 7231

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 2820	. . 3 ⊢ ∅ = ∅
2	1	orci 861	. 2 ⊢ (∅ = ∅ ∨ 𝐵 = ∅)
3		0mpo0 7230	. 2 ⊢ ((∅ = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)
4	2, 3	ax-mp 5	1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Syntax hints:   ∨ wo 843   = wceq 1536  ∅c0 4284   ∈ cmpo 7151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-dif 3932  df-nul 4285  df-oprab 7153  df-mpo 7154

This theorem is referenced by: (None)