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Mirrors > Home > MPE Home > Th. List > ndmovg | Structured version Visualization version GIF version |
Description: The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.) |
Ref | Expression |
---|---|
ndmovg | ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6653 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | eleq2 2690 | . . . . . 6 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆))) | |
3 | opelxp 5146 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
4 | 2, 3 | syl6bb 276 | . . . . 5 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
5 | 4 | notbid 308 | . . . 4 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
6 | ndmfv 6218 | . . . 4 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
7 | 5, 6 | syl6bir 244 | . . 3 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐹‘〈𝐴, 𝐵〉) = ∅)) |
8 | 7 | imp 445 | . 2 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
9 | 1, 8 | syl5eq 2668 | 1 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 〈cop 4183 × cxp 5112 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: ndmov 6818 curry1val 7270 curry2val 7274 1div0 10686 repsundef 13518 cshnz 13538 mamufacex 20195 mavmulsolcl 20357 mavmul0g 20359 iscau2 23075 1div0apr 27324 |
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