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Theorem mpt2xopynvov0 7304
 Description: If the second 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 element of the first component of the first argument, 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
mpt2xopynvov0 (𝐾𝑉 → (⟨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 mpt2xopynvov0
StepHypRef Expression
1 mpt2xopn0yelv.f . . . 4 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
21mpt2xopynvov0g 7300 . . 3 (((𝑉 ∈ V ∧ 𝑊 ∈ V) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
32ex 450 . 2 ((𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅))
41mpt2xopxprcov0 7303 . . 3 (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
54a1d 25 . 2 (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅))
63, 5pm2.61i 176 1 (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ∉ wnel 2893  Vcvv 3190  ∅c0 3897  ⟨cop 4161  ‘cfv 5857  (class class class)co 6615   ↦ cmpt2 6617  1st c1st 7126 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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 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-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129 This theorem is referenced by:  mpt2xopoveqd  7307
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