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Theorem ndmovcl 6717
 Description: The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)
Hypotheses
Ref Expression
ndmov.1 dom 𝐹 = (𝑆 × 𝑆)
ndmovcl.2 ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
ndmovcl.3 ∅ ∈ 𝑆
Assertion
Ref Expression
ndmovcl (𝐴𝐹𝐵) ∈ 𝑆

Proof of Theorem ndmovcl
StepHypRef Expression
1 ndmovcl.2 . 2 ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
2 ndmov.1 . . . 4 dom 𝐹 = (𝑆 × 𝑆)
32ndmov 6716 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
4 ndmovcl.3 . . 3 ∅ ∈ 𝑆
53, 4syl6eqel 2696 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
61, 5pm2.61i 175 1 (𝐴𝐹𝐵) ∈ 𝑆
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874   × cxp 5036  dom cdm 5038  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by: (None)
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