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Theorem mpondm0 7649
Description: The value of an operation given by a maps-to rule is the empty set if the arguments are not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Hypothesis
Ref Expression
mpondm0.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
Assertion
Ref Expression
mpondm0 (¬ (𝑉𝑋𝑊𝑌) → (𝑉𝐹𝑊) = ∅)
Distinct variable groups:   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mpondm0
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mpondm0.f . . . . 5 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
2 df-mpo 7416 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2760 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
43dmeqi 5904 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
5 dmoprabss 7513 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
64, 5eqsstri 4016 . 2 dom 𝐹 ⊆ (𝑋 × 𝑌)
7 nssdmovg 7591 . 2 ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉𝑋𝑊𝑌)) → (𝑉𝐹𝑊) = ∅)
86, 7mpan 688 1 (¬ (𝑉𝑋𝑊𝑌) → (𝑉𝐹𝑊) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wss 3948  c0 4322   × cxp 5674  dom cdm 5676  (class class class)co 7411  {coprab 7412  cmpo 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  2mpo0  7657  elovmpt3imp  7665  el2mpocsbcl  8073  bropopvvv  8078  supp0prc  8151  brovex  8209  swrdnznd  14596  pfxnndmnd  14626  fullfunc  17861  fthfunc  17862  natfval  17901  evlval  21877  matbas0  22130  matrcl  22132  marrepfval  22282  marepvfval  22287  submafval  22301  minmar1fval  22368  hmeofval  23482  nghmfval  24459  wspthsn  29357  iswwlksnon  29362  iswspthsnon  29365  clwwlkn  29534  clwwlkneq0  29537  clwwlknon  29598  clwwlk0on0  29600  clwwlknon0  29601  naryfval  47402  naryfvalixp  47403
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