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Theorem mpondm0 7673
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 7436 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2763 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
43dmeqi 5918 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
5 dmoprabss 7536 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
64, 5eqsstri 4030 . 2 dom 𝐹 ⊆ (𝑋 × 𝑌)
7 nssdmovg 7615 . 2 ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉𝑋𝑊𝑌)) → (𝑉𝐹𝑊) = ∅)
86, 7mpan 690 1 (¬ (𝑉𝑋𝑊𝑌) → (𝑉𝐹𝑊) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963  c0 4339   × cxp 5687  dom cdm 5689  (class class class)co 7431  {coprab 7432  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  2mpo0  7682  elovmpt3imp  7690  el2mpocsbcl  8109  bropopvvv  8114  supp0prc  8187  brovex  8246  swrdnznd  14677  pfxnndmnd  14707  fullfunc  17960  fthfunc  17961  natfval  18001  evlval  22137  matbas0  22430  matrcl  22432  marrepfval  22582  marepvfval  22587  submafval  22601  minmar1fval  22668  hmeofval  23782  nghmfval  24759  wspthsn  29878  iswwlksnon  29883  iswspthsnon  29886  clwwlkn  30055  clwwlkneq0  30058  clwwlknon  30119  clwwlk0on0  30121  clwwlknon0  30122  naryfval  48478  naryfvalixp  48479
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