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Theorem ndmovcl 7594
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 7593 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
4 ndmovcl.3 . . 3 ∅ ∈ 𝑆
53, 4eqeltrdi 2839 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
61, 5pm2.61i 182 1 (𝐴𝐹𝐵) ∈ 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wcel 2104  c0 4321   × cxp 5673  dom cdm 5675  (class class class)co 7411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-dm 5685  df-iota 6494  df-fv 6550  df-ov 7414
This theorem is referenced by: (None)
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