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Theorem ndmovcl 7322
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 7321 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
4 ndmovcl.3 . . 3 ∅ ∈ 𝑆
53, 4syl6eqel 2918 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
61, 5pm2.61i 183 1 (𝐴𝐹𝐵) ∈ 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  c0 4288   × cxp 5546  dom cdm 5548  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by: (None)
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