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Theorem ndmovcl 7587
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 7586 . . 3 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
4 ndmovcl.3 . . 3 ∅ ∈ 𝑆
53, 4eqeltrdi 2842 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
61, 5pm2.61i 182 1 (𝐴𝐹𝐵) ∈ 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  c0 4321   × cxp 5673  dom cdm 5675  (class class class)co 7404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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 6492  df-fv 6548  df-ov 7407
This theorem is referenced by: (None)
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