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 2870	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
6	1, 5	pm2.61i 184	1 ⊢ (𝐴𝐹𝐵) ∈ 𝑆

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∅c0 4285   × cxp 5658  dom cdm 5660  (class class class)co 7412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-dm 5670  df-iota 6492  df-fv 6544  df-ov 7415

This theorem is used by: (None)