Theorem mpoxopn0yelv 8185

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 8039	. . . 4 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))
3		elfvdm 6918	. . . . 5 ⊢ (𝑁 ∈ (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
4		df-ov 7399	. . . . 5 ⊢ (⟨𝑉, 𝑊⟩𝐹𝐾) = (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)
5	3, 4	eleq2s 2852	. . . 4 ⊢ (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
6	2, 5	sselid 3978	. . 3 ⊢ (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)))
7		fveq2 6881	. . . . 5 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → (1st ‘𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
8	7	opeliunxp2 5833	. . . 4 ⊢ (⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (⟨𝑉, 𝑊⟩ ∈ V ∧ 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩)))
9	8	simprbi 498	. . 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 7974	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
12	11	eleq2d 2820	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩) ↔ 𝐾 ∈ 𝑉))
13	10, 12	imbitrid 243	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4624  ⟨cop 4630  ∪ ciun 4993   × cxp 5670  dom cdm 5672  ‘cfv 6535  (class class class)co 7396   ∈ cmpo 7398  1^st c1st 7960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7962  df-2nd 7963

This theorem is referenced by:  mpoxopynvov0g  8186  mpoxopovel  8192