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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaovg | Structured version Visualization version GIF version |
Description: The value of an operation outside its domain, analogous to ndmovg 6715. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaovg | ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5070 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
2 | eleq2 2677 | . . . . . 6 ⊢ ((𝑅 × 𝑆) = dom 𝐹 → (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) | |
3 | 2 | eqcoms 2618 | . . . . 5 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) |
4 | 1, 3 | syl5bbr 273 | . . . 4 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) |
5 | 4 | notbid 307 | . . 3 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹)) |
6 | 5 | biimpa 500 | . 2 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
7 | ndmaov 39912 | . 2 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | |
8 | 6, 7 | syl 17 | 1 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 dom cdm 5038 ((caov 39844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-fv 5812 df-dfat 39845 df-afv 39846 df-aov 39847 |
This theorem is referenced by: (None) |
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