Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2xopxprcov0 Structured version   Visualization version   GIF version

Theorem mpt2xopxprcov0 7288
 Description: If the components of 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, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopn0yelv.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
Assertion
Ref Expression
mpt2xopxprcov0 (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐾   𝑥,𝑉   𝑥,𝑊   𝑥,𝐹
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑦)   𝐾(𝑦)   𝑉(𝑦)   𝑊(𝑦)

Proof of Theorem mpt2xopxprcov0
StepHypRef Expression
1 opelxp 5106 . 2 (⟨𝑉, 𝑊⟩ ∈ (V × V) ↔ (𝑉 ∈ V ∧ 𝑊 ∈ V))
2 mpt2xopn0yelv.f . . 3 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
32mpt2xopxnop0 7286 . 2 (¬ ⟨𝑉, 𝑊⟩ ∈ (V × V) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
41, 3sylnbir 321 1 (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ∅c0 3891  ⟨cop 4154   × cxp 5072  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  1st c1st 7111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114 This theorem is referenced by:  mpt2xopynvov0  7289
 Copyright terms: Public domain W3C validator