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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 7321 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
4 | ndmovcl.3 | . . 3 ⊢ ∅ ∈ 𝑆 | |
5 | 3, 4 | syl6eqel 2918 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
6 | 1, 5 | pm2.61i 183 | 1 ⊢ (𝐴𝐹𝐵) ∈ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 × cxp 5546 dom cdm 5548 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: (None) |
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