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Theorem mpt2ndm0 6840
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
mpt2ndm0.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
Assertion
Ref Expression
mpt2ndm0 (¬ (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 mpt2ndm0
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mpt2ndm0.f . . . . 5 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
2 df-mpt2 6620 . . . . 5 (𝑥𝑋, 𝑦𝑌𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2643 . . . 4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
43dmeqi 5295 . . 3 dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)}
5 dmoprabss 6707 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑋𝑦𝑌) ∧ 𝑧 = 𝐶)} ⊆ (𝑋 × 𝑌)
64, 5eqsstri 3620 . 2 dom 𝐹 ⊆ (𝑋 × 𝑌)
7 nssdmovg 6781 . 2 ((dom 𝐹 ⊆ (𝑋 × 𝑌) ∧ ¬ (𝑉𝑋𝑊𝑌)) → (𝑉𝐹𝑊) = ∅)
86, 7mpan 705 1 (¬ (𝑉𝑋𝑊𝑌) → (𝑉𝐹𝑊) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3560  c0 3897   × cxp 5082  dom cdm 5084  (class class class)co 6615  {coprab 6616  cmpt2 6617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-dm 5094  df-iota 5820  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  2mpt20  6847  elovmpt3imp  6855  el2mpt2csbcl  7210  bropopvvv  7215  supp0prc  7258  brovex  7308  fullfunc  16506  fthfunc  16507  natfval  16546  evlval  19464  matbas0  20156  matrcl  20158  marrepfval  20306  marepvfval  20311  submafval  20325  minmar1fval  20392  hmeofval  21501  nghmfval  22466  wspthsn  26638  iswwlksnon  26643  iswspthsnon  26644  wspn0  26723  numclwwlkffin0  27105
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