Theorem mpoxopynvov0g 8221

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 ∈ (1st ‘𝑥) ↦ 𝐶)

Assertion

Ref	Expression
mpoxopynvov0g	⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∉ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)

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.

Step	Hyp	Ref	Expression
1		neq0 4332	. . . 4 ⊢ (¬ (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅ ↔ ∃𝑛 𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾))
2		mpoxopn0yelv.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
3	2	mpoxopn0yelv 8220	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
4		nnel 3045	. . . . . 6 ⊢ (¬ 𝐾 ∉ 𝑉 ↔ 𝐾 ∈ 𝑉)
5	3, 4	imbitrrdi 252	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ¬ 𝐾 ∉ 𝑉))
6	5	exlimdv 1932	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (∃𝑛 𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ¬ 𝐾 ∉ 𝑉))
7	1, 6	biimtrid 242	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (¬ (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅ → ¬ 𝐾 ∉ 𝑉))
8	7	con4d 115	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅))
9	8	imp 406	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∉ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ∉ wnel 3035  Vcvv 3463  ∅c0 4313  ⟨cop 4612  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997

This theorem is referenced by:  mpoxopynvov0  8225