MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpoxopxnop0 Structured version   Visualization version   GIF version

Theorem mpoxopxnop0 8199
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 4345 . . 3 (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾))
2 mpoxopn0yelv.f . . . . . . 7 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
32dmmpossx 8051 . . . . . 6 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
4 elfvdm 6928 . . . . . . 7 (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
5 df-ov 7411 . . . . . . 7 (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩)
64, 5eleq2s 2851 . . . . . 6 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
73, 6sselid 3980 . . . . 5 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
8 fveq2 6891 . . . . . . 7 (𝑥 = 𝑉 → (1st𝑥) = (1st𝑉))
98opeliunxp2 5838 . . . . . 6 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)))
10 eluni 4911 . . . . . . . . 9 (𝐾 dom {𝑉} ↔ ∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}))
11 ne0i 4334 . . . . . . . . . . . . 13 (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅)
1211ad2antlr 725 . . . . . . . . . . . 12 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅)
13 dmsnn0 6206 . . . . . . . . . . . 12 (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅)
1412, 13sylibr 233 . . . . . . . . . . 11 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V))
1514ex 413 . . . . . . . . . 10 ((𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1615exlimiv 1933 . . . . . . . . 9 (∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1710, 16sylbi 216 . . . . . . . 8 (𝐾 dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
18 1stval 7976 . . . . . . . 8 (1st𝑉) = dom {𝑉}
1917, 18eleq2s 2851 . . . . . . 7 (𝐾 ∈ (1st𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
2019impcom 408 . . . . . 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 1933 . . 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 396   = wceq 1541  wex 1781  wcel 2106  wne 2940  Vcvv 3474  c0 4322  {csn 4628  cop 4634   cuni 4908   ciun 4997   × cxp 5674  dom cdm 5676  cfv 6543  (class class class)co 7408  cmpo 7410  1st c1st 7972
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975
This theorem is referenced by:  mpoxopx0ov0  8200  mpoxopxprcov0  8201
  Copyright terms: Public domain W3C validator