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Mirrors > Home > MPE Home > Th. List > ndmovcl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ndmov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
ndmovcl.2 | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
ndmovcl.3 | ⊢ ∅ ∈ 𝑆 |
Ref | Expression |
---|---|
ndmovcl | ⊢ (𝐴𝐹𝐵) ∈ 𝑆 |
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 7593 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
4 | ndmovcl.3 | . . 3 ⊢ ∅ ∈ 𝑆 | |
5 | 3, 4 | eqeltrdi 2839 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
6 | 1, 5 | pm2.61i 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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