Theorem mpo0 7518

Description: A mapping operation with empty domain. In this version of mpo0v 7517, the class of the second operator may depend on the first operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
mpo0	⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Proof of Theorem mpo0

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 7436	. 2 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		df-oprab 7435	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
3		noel 4344	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
4		simprll 779	. . . . . . 7 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅)
5	3, 4	mto 197	. . . . . 6 ⊢ ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
6	5	nex 1797	. . . . 5 ⊢ ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
7	6	nex 1797	. . . 4 ⊢ ¬ ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
8	7	nex 1797	. . 3 ⊢ ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
9	8	abf 4412	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = ∅
10	1, 2, 9	3eqtri 2767	1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712  ∅c0 4339  ⟨cop 4637  {coprab 7432   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-nul 4340  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  coafval  18118  d0mat2pmat  22760