Theorem mpoxopynvov0g 8189

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 4304	. . . 4 ⊢ (¬ (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅ ↔ ∃𝑛 𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾))
2		mpoxopn0yelv.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
3	2	mpoxopn0yelv 8188	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
4		nnel 3070	. . . . . 6 ⊢ (¬ 𝐾 ∉ 𝑉 ↔ 𝐾 ∈ 𝑉)
5	3, 4	imbitrrdi 254	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ¬ 𝐾 ∉ 𝑉))
6	5	exlimdv 1952	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (∃𝑛 𝑛 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ¬ 𝐾 ∉ 𝑉))
7	1, 6	biimtrid 244	. . 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 410	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∉ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ∉ wnel 3060  Vcvv 3453  ∅c0 4285  ⟨cop 4587  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  1^st c1st 7964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967

This theorem is referenced by:  mpoxopynvov0  8193