MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpoxneldm Structured version   Visualization version   GIF version

Theorem mpoxneldm 8237
Description: If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)
Hypothesis
Ref Expression
mpoxeldm.f 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
mpoxneldm ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
Distinct variable groups:   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐷(𝑥)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem mpoxneldm
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 df-nel 3047 . . . 4 (𝑋𝐶 ↔ ¬ 𝑋𝐶)
2 df-nel 3047 . . . 4 (𝑌𝑋 / 𝑥𝐷 ↔ ¬ 𝑌𝑋 / 𝑥𝐷)
31, 2orbi12i 915 . . 3 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
4 ianor 984 . . 3 (¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
53, 4bitr4i 278 . 2 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ ¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
6 neq0 4352 . . . 4 (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7 mpoxeldm.f . . . . . 6 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
87mpoxeldm 8236 . . . . 5 (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
98exlimiv 1930 . . . 4 (∃𝑛 𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
106, 9sylbi 217 . . 3 (¬ (𝑋𝐹𝑌) = ∅ → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
1110con1i 147 . 2 (¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
125, 11sylbi 217 1 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wnel 3046  csb 3899  c0 4333  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015
This theorem is referenced by:  nbgrnvtx0  29356  clnbgrnvtx0  47814
  Copyright terms: Public domain W3C validator