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Theorem mpt20 5601
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
mpt20 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅

Proof of Theorem mpt20
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt2 5544 . 2 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-oprab 5543 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))}
3 noel 3255 . . . . . . 7 ¬ 𝑥 ∈ ∅
4 simprll 497 . . . . . . 7 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅)
53, 4mto 598 . . . . . 6 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
65nex 1405 . . . . 5 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
76nex 1405 . . . 4 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
87nex 1405 . . 3 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
98abf 3287 . 2 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))} = ∅
101, 2, 93eqtri 2080 1 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  c0 3251  cop 3405  {coprab 5540  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-in 2951  df-ss 2958  df-nul 3252  df-oprab 5543  df-mpt2 5544
This theorem is referenced by: (None)
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