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Theorem ndmovcl 7335
 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 7334 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
4 ndmovcl.3 . . 3 ∅ ∈ 𝑆
53, 4eqeltrdi 2860 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
61, 5pm2.61i 185 1 (𝐴𝐹𝐵) ∈ 𝑆
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4227   × cxp 5526  dom cdm 5528  (class class class)co 7156 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-xp 5534  df-dm 5538  df-iota 6299  df-fv 6348  df-ov 7159 This theorem is referenced by: (None)
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