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Theorem mpoxopn0yelv 7911
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 ∈ (1st𝑥) ↦ 𝐶)
Assertion
Ref Expression
mpoxopn0yelv ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
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
StepHypRef Expression
1 mpoxopn0yelv.f . . . . 5 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
21dmmpossx 7792 . . . 4 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
3 elfvdm 6709 . . . . 5 (𝑁 ∈ (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
4 df-ov 7176 . . . . 5 (⟨𝑉, 𝑊𝐹𝐾) = (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)
53, 4eleq2s 2852 . . . 4 (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
62, 5sseldi 3876 . . 3 (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
7 fveq2 6677 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
87opeliunxp2 5682 . . . 4 (⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) ↔ (⟨𝑉, 𝑊⟩ ∈ V ∧ 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩)))
98simprbi 500 . . 3 (⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) → 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩))
106, 9syl 17 . 2 (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩))
11 op1stg 7729 . . 3 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
1211eleq2d 2819 . 2 ((𝑉𝑋𝑊𝑌) → (𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩) ↔ 𝐾𝑉))
1310, 12syl5ib 247 1 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3399  {csn 4517  cop 4523   ciun 4882   × cxp 5524  dom cdm 5526  cfv 6340  (class class class)co 7173  cmpo 7175  1st c1st 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-1st 7717  df-2nd 7718
This theorem is referenced by:  mpoxopynvov0g  7912  mpoxopovel  7918
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