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Theorem mpoxopynvov0g 7881
 Description: If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
Hypothesis
Ref Expression
mpoxopn0yelv.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
Assertion
Ref Expression
mpoxopynvov0g (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐾   𝑥,𝑉   𝑥,𝑊
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐾(𝑦)   𝑉(𝑦)   𝑊(𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem mpoxopynvov0g
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 neq0 4262 . . . 4 (¬ (⟨𝑉, 𝑊𝐹𝐾) = ∅ ↔ ∃𝑛 𝑛 ∈ (⟨𝑉, 𝑊𝐹𝐾))
2 mpoxopn0yelv.f . . . . . . 7 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
32mpoxopn0yelv 7880 . . . . . 6 ((𝑉𝑋𝑊𝑌) → (𝑛 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
4 nnel 3100 . . . . . 6 𝐾𝑉𝐾𝑉)
53, 4syl6ibr 255 . . . . 5 ((𝑉𝑋𝑊𝑌) → (𝑛 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ¬ 𝐾𝑉))
65exlimdv 1934 . . . 4 ((𝑉𝑋𝑊𝑌) → (∃𝑛 𝑛 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ¬ 𝐾𝑉))
71, 6syl5bi 245 . . 3 ((𝑉𝑋𝑊𝑌) → (¬ (⟨𝑉, 𝑊𝐹𝐾) = ∅ → ¬ 𝐾𝑉))
87con4d 115 . 2 ((𝑉𝑋𝑊𝑌) → (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅))
98imp 410 1 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ∉ wnel 3091  Vcvv 3442  ∅c0 4246  ⟨cop 4534  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147  1st c1st 7682 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685 This theorem is referenced by:  mpoxopynvov0  7885
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