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Theorem mpondm0 7647
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 7414 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2761 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
43dmeqi 5905 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
5 dmoprabss 7511 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
64, 5eqsstri 4017 . 2 dom 𝐹 ⊆ (𝑋 × 𝑌)
7 nssdmovg 7589 . 2 ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉𝑋𝑊𝑌)) → (𝑉𝐹𝑊) = ∅)
86, 7mpan 689 1 (¬ (𝑉𝑋𝑊𝑌) → (𝑉𝐹𝑊) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3949  c0 4323   × cxp 5675  dom cdm 5677  (class class class)co 7409  {coprab 7410  cmpo 7411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414
This theorem is referenced by:  2mpo0  7655  elovmpt3imp  7663  el2mpocsbcl  8071  bropopvvv  8076  supp0prc  8149  brovex  8207  swrdnznd  14592  pfxnndmnd  14622  fullfunc  17857  fthfunc  17858  natfval  17897  evlval  21658  matbas0  21910  matrcl  21912  marrepfval  22062  marepvfval  22067  submafval  22081  minmar1fval  22148  hmeofval  23262  nghmfval  24239  wspthsn  29102  iswwlksnon  29107  iswspthsnon  29110  clwwlkn  29279  clwwlkneq0  29282  clwwlknon  29343  clwwlk0on0  29345  clwwlknon0  29346  naryfval  47314  naryfvalixp  47315
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