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Theorem mpoxopn0yelv 7862
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 7747 . . . 4 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
3 elfvdm 6683 . . . . 5 (𝑁 ∈ (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
4 df-ov 7141 . . . . 5 (⟨𝑉, 𝑊𝐹𝐾) = (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)
53, 4eleq2s 2934 . . . 4 (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
62, 5sseldi 3949 . . 3 (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
7 fveq2 6651 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
87opeliunxp2 5690 . . . 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 7684 . . 3 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
1211eleq2d 2901 . 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 1538  wcel 2115  Vcvv 3479  {csn 4548  cop 4554   ciun 4900   × cxp 5534  dom cdm 5536  cfv 6336  (class class class)co 7138  cmpo 7140  1st c1st 7670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673
This theorem is referenced by:  mpoxopynvov0g  7863  mpoxopovel  7869
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