Theorem ndmovcl 7630

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 7629	. . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅)
4		ndmovcl.3	. . 3 ⊢ ∅ ∈ 𝑆
5	3, 4	eqeltrdi 2872	. 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 401   = wceq 1570   ∈ wcel 2147  ∅c0 4281   × cxp 5653  dom cdm 5655  (class class class)co 7443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-dm 5665  df-iota 6512  df-fv 6566  df-ov 7446

This theorem is used by: (None)