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Theorem mpt2xopn0yelv 7604
 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
mpt2xopn0yelv.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
Assertion
Ref Expression
mpt2xopn0yelv ((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 mpt2xopn0yelv
StepHypRef Expression
1 mpt2xopn0yelv.f . . . . 5 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
21dmmpt2ssx 7498 . . . 4 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
3 elfvdm 6465 . . . . 5 (𝑁 ∈ (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
4 df-ov 6908 . . . . 5 (⟨𝑉, 𝑊𝐹𝐾) = (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)
53, 4eleq2s 2924 . . . 4 (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ dom 𝐹)
62, 5sseldi 3825 . . 3 (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
7 fveq2 6433 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
87opeliunxp2 5493 . . . 4 (⟨⟨𝑉, 𝑊⟩, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) ↔ (⟨𝑉, 𝑊⟩ ∈ V ∧ 𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩)))
98simprbi 492 . . 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 7440 . . 3 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
1211eleq2d 2892 . 2 ((𝑉𝑋𝑊𝑌) → (𝐾 ∈ (1st ‘⟨𝑉, 𝑊⟩) ↔ 𝐾𝑉))
1310, 12syl5ib 236 1 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3414  {csn 4397  ⟨cop 4403  ∪ ciun 4740   × cxp 5340  dom cdm 5342  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907  1st c1st 7426 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429 This theorem is referenced by:  mpt2xopynvov0g  7605  mpt2xopovel  7611
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