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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 7586 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
4 | ndmovcl.3 | . . 3 ⊢ ∅ ∈ 𝑆 | |
5 | 3, 4 | eqeltrdi 2842 | . 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 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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