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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 7145 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | eleq2 2901 | . . . . . 6 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆))) | |
3 | opelxp 5577 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
4 | 2, 3 | syl6bb 289 | . . . . 5 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
5 | 4 | notbid 320 | . . . 4 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 ↔ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
6 | ndmfv 6686 | . . . 4 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
7 | 5, 6 | syl6bir 256 | . . 3 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐹‘〈𝐴, 𝐵〉) = ∅)) |
8 | 7 | imp 409 | . 2 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
9 | 1, 8 | syl5eq 2868 | 1 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4279 〈cop 4559 × cxp 5539 dom cdm 5541 ‘cfv 6341 (class class class)co 7142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-xp 5547 df-dm 5551 df-iota 6300 df-fv 6349 df-ov 7145 |
This theorem is referenced by: ndmov 7318 curry1val 7786 curry2val 7790 1div0 11285 repsundef 14118 cshnz 14139 mamufacex 20983 mavmulsolcl 21143 mavmul0g 21145 iscau2 23863 1div0apr 28231 rrxsphere 44820 |
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