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Theorem mpondm0 7446
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 7218 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2765 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
43dmeqi 5773 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
5 dmoprabss 7313 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
64, 5eqsstri 3935 . 2 dom 𝐹 ⊆ (𝑋 × 𝑌)
7 nssdmovg 7390 . 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 399   = wceq 1543  wcel 2110  wss 3866  c0 4237   × cxp 5549  dom cdm 5551  (class class class)co 7213  {coprab 7214  cmpo 7215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-dm 5561  df-iota 6338  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218
This theorem is referenced by:  2mpo0  7454  elovmpt3imp  7462  el2mpocsbcl  7853  bropopvvv  7858  supp0prc  7906  brovex  7964  swrdnznd  14207  pfxnndmnd  14237  fullfunc  17413  fthfunc  17414  natfval  17453  evlval  21055  matbas0  21307  matrcl  21309  marrepfval  21457  marepvfval  21462  submafval  21476  minmar1fval  21543  hmeofval  22655  nghmfval  23620  wspthsn  27932  iswwlksnon  27937  iswspthsnon  27940  clwwlkn  28109  clwwlkneq0  28112  clwwlknon  28173  clwwlk0on0  28175  clwwlknon0  28176  naryfval  45647  naryfvalixp  45648
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