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Theorem mpo0 5939
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 ∈ ∅, 𝑦𝐵𝐶) = ∅

Proof of Theorem mpo0
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpo 5874 . 2 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-oprab 5873 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))}
3 noel 3426 . . . . . . 7 ¬ 𝑥 ∈ ∅
4 simprll 537 . . . . . . 7 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅)
53, 4mto 662 . . . . . 6 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
65nex 1500 . . . . 5 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
76nex 1500 . . . 4 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
87nex 1500 . . 3 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
98abf 3466 . 2 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))} = ∅
101, 2, 93eqtri 2202 1 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  {cab 2163  c0 3422  cop 3594  {coprab 5870  cmpo 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-oprab 5873  df-mpo 5874
This theorem is referenced by: (None)
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