Theorem ndmovcl 7597

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	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
ndmovcl.3	⊢ ∅ ∈ 𝑆

Assertion

Ref	Expression
ndmovcl	⊢ (𝐴𝐹𝐵) ∈ 𝑆

Proof of Theorem ndmovcl

Step	Hyp	Ref	Expression
1		ndmovcl.2	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
2		ndmov.1	. . . 4 ⊢ dom 𝐹 = (𝑆 × 𝑆)
3	2	ndmov 7596	. . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅)
4		ndmovcl.3	. . 3 ⊢ ∅ ∈ 𝑆
5	3, 4	eqeltrdi 2843	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
6	1, 5	pm2.61i 182	1 ⊢ (𝐴𝐹𝐵) ∈ 𝑆

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∅c0 4313   × cxp 5657  dom cdm 5659  (class class class)co 7410

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-dm 5669  df-iota 6489  df-fv 6544  df-ov 7413

This theorem is referenced by: (None)