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Theorem mpt20 6923
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 6847 . 2 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-oprab 6846 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))}
3 noel 4083 . . . . . . 7 ¬ 𝑥 ∈ ∅
4 simprll 797 . . . . . . 7 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅)
53, 4mto 188 . . . . . 6 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
65nex 1895 . . . . 5 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
76nex 1895 . . . 4 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
87nex 1895 . . 3 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
98abf 4140 . 2 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))} = ∅
101, 2, 93eqtri 2791 1 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  c0 4079  cop 4340  {coprab 6843  cmpt2 6844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-nul 4080  df-oprab 6846  df-mpt2 6847
This theorem is referenced by:  homffval  16617  comfffval  16625  natfval  16873  coafval  16981  xpchomfval  17087  xpccofval  17090  plusffval  17515  grpsubfval  17733  oppglsm  18323  dvrfval  18951  scaffval  19150  psrmulr  19658  ipffval  20268  marrepfval  20643  marepvfval  20648  d0mat2pmat  20822  pcofval  23088  clwwlknonmpt2  27317  mendplusgfval  38364  mendmulrfval  38366  mendvscafval  38369
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