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Theorem mpt20 6690
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 6620 . 2 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-oprab 6619 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))}
3 noel 3901 . . . . . . 7 ¬ 𝑥 ∈ ∅
4 simprll 801 . . . . . . 7 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅)
53, 4mto 188 . . . . . 6 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
65nex 1728 . . . . 5 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
76nex 1728 . . . 4 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
87nex 1728 . . 3 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
98abf 3956 . 2 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))} = ∅
101, 2, 93eqtri 2647 1 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  c0 3897  cop 4161  {coprab 6616  cmpt2 6617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-nul 3898  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  homffval  16290  comfffval  16298  natfval  16546  coafval  16654  xpchomfval  16759  xpccofval  16762  plusffval  17187  grpsubfval  17404  oppglsm  17997  dvrfval  18624  scaffval  18821  psrmulr  19324  ipffval  19933  marrepfval  20306  marepvfval  20311  d0mat2pmat  20483  pcofval  22750  mendplusgfval  37275  mendmulrfval  37277  mendvscafval  37280
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