Theorem mpoxopn0yelv 8237

Description: If there is an element of the value 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, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Hypothesis

Ref	Expression
mpoxopn0yelv.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)

Assertion

Ref	Expression
mpoxopn0yelv	⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))

Distinct variable groups:   𝑥,𝑦   𝑥,𝐾   𝑥,𝑉   𝑥,𝑊

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐾(𝑦)   𝑁(𝑥,𝑦)   𝑉(𝑦)   𝑊(𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem mpoxopn0yelv

Step	Hyp	Ref	Expression
1		mpoxopn0yelv.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
2	1	dmmpossx 8090	. . . 4 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))
3		elfvdm 6944	. . . . 5 ⊢ (𝑁 ∈ (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
4		df-ov 7434	. . . . 5 ⊢ (⟨𝑉, 𝑊⟩𝐹𝐾) = (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)
5	3, 4	eleq2s 2857	. . . 4 ⊢ (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
6	2, 5	sselid 3993	. . 3 ⊢ (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)))
7		fveq2 6907	. . . . 5 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → (1st ‘𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
8	7	opeliunxp2 5852	. . . 4 ⊢ (⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (⟨𝑉, 𝑊⟩ ∈ V ∧ 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩)))
9	8	simprbi 496	. . 3 ⊢ (⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) → 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩))
10	6, 9	syl 17	. 2 ⊢ (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩))
11		op1stg 8025	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
12	11	eleq2d 2825	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩) ↔ 𝐾 ∈ 𝑉))
13	10, 12	imbitrid 244	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ⟨cop 4637  ∪ ciun 4996   × cxp 5687  dom cdm 5689  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014

This theorem is referenced by:  mpoxopynvov0g  8238  mpoxopovel  8244