MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt20 Structured version   Visualization version   GIF version

Theorem mpt20 6871
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 6797 . 2 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-oprab 6796 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))}
3 noel 4065 . . . . . . 7 ¬ 𝑥 ∈ ∅
4 simprll 756 . . . . . . 7 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶)) → 𝑥 ∈ ∅)
53, 4mto 188 . . . . . 6 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
65nex 1878 . . . . 5 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
76nex 1878 . . . 4 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
87nex 1878 . . 3 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))
98abf 4120 . 2 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ((𝑥 ∈ ∅ ∧ 𝑦𝐵) ∧ 𝑧 = 𝐶))} = ∅
101, 2, 93eqtri 2796 1 (𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1630  wex 1851  wcel 2144  {cab 2756  c0 4061  cop 4320  {coprab 6793  cmpt2 6794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-nul 4062  df-oprab 6796  df-mpt2 6797
This theorem is referenced by:  homffval  16556  comfffval  16564  natfval  16812  coafval  16920  xpchomfval  17026  xpccofval  17029  plusffval  17454  grpsubfval  17671  oppglsm  18263  dvrfval  18891  scaffval  19090  psrmulr  19598  ipffval  20209  marrepfval  20583  marepvfval  20588  d0mat2pmat  20762  pcofval  23028  clwwlknonmpt2  27258  mendplusgfval  38274  mendmulrfval  38276  mendvscafval  38279
  Copyright terms: Public domain W3C validator