Theorem mpo0 5849

Description: A mapping operation with empty domain. (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 5787	. 2 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		df-oprab 5786	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
3		noel 3372	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
4		simprll 527	. . . . . . 7 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅)
5	3, 4	mto 652	. . . . . 6 ⊢ ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
6	5	nex 1477	. . . . 5 ⊢ ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
7	6	nex 1477	. . . 4 ⊢ ¬ ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
8	7	nex 1477	. . 3 ⊢ ¬ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
9	8	abf 3411	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = ∅
10	1, 2, 9	3eqtri 2165	1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅

Syntax hints:   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∅c0 3368  ⟨cop 3535  {coprab 5783   ∈ cmpo 5784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-oprab 5786  df-mpo 5787

This theorem is referenced by: (None)