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Theorem mpoxopxnop0 8195
Description: If the first argument 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, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
Hypothesis
Ref Expression
mpoxopn0yelv.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
Assertion
Ref Expression
mpoxopxnop0 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐾   𝑥,𝑉   𝑥,𝐹
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑦)   𝐾(𝑦)   𝑉(𝑦)

Proof of Theorem mpoxopxnop0
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 neq0 4344 . . 3 (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾))
2 mpoxopn0yelv.f . . . . . . 7 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
32dmmpossx 8047 . . . . . 6 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
4 elfvdm 6925 . . . . . . 7 (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
5 df-ov 7407 . . . . . . 7 (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩)
64, 5eleq2s 2852 . . . . . 6 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
73, 6sselid 3979 . . . . 5 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
8 fveq2 6888 . . . . . . 7 (𝑥 = 𝑉 → (1st𝑥) = (1st𝑉))
98opeliunxp2 5836 . . . . . 6 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)))
10 eluni 4910 . . . . . . . . 9 (𝐾 dom {𝑉} ↔ ∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}))
11 ne0i 4333 . . . . . . . . . . . . 13 (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅)
1211ad2antlr 726 . . . . . . . . . . . 12 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅)
13 dmsnn0 6203 . . . . . . . . . . . 12 (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅)
1412, 13sylibr 233 . . . . . . . . . . 11 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V))
1514ex 414 . . . . . . . . . 10 ((𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1615exlimiv 1934 . . . . . . . . 9 (∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1710, 16sylbi 216 . . . . . . . 8 (𝐾 dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
18 1stval 7972 . . . . . . . 8 (1st𝑉) = dom {𝑉}
1917, 18eleq2s 2852 . . . . . . 7 (𝐾 ∈ (1st𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
2019impcom 409 . . . . . 6 ((𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)) → 𝑉 ∈ (V × V))
219, 20sylbi 216 . . . . 5 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) → 𝑉 ∈ (V × V))
227, 21syl 17 . . . 4 (𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
2322exlimiv 1934 . . 3 (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
241, 23sylbi 216 . 2 (¬ (𝑉𝐹𝐾) = ∅ → 𝑉 ∈ (V × V))
2524con1i 147 1 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  Vcvv 3475  c0 4321  {csn 4627  cop 4633   cuni 4907   ciun 4996   × cxp 5673  dom cdm 5675  cfv 6540  (class class class)co 7404  cmpo 7406  1st c1st 7968
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 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971
This theorem is referenced by:  mpoxopx0ov0  8196  mpoxopxprcov0  8197
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