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Mirrors > Home > MPE Home > Th. List > nssdmovg | Structured version Visualization version GIF version |
Description: The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.) |
Ref | Expression |
---|---|
nssdmovg | ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7216 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | ssel2 3895 | . . . . 5 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
3 | opelxp 5587 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
4 | 2, 3 | sylib 221 | . . . 4 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) |
5 | 4 | stoic1a 1780 | . . 3 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
6 | ndmfv 6747 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
7 | 5, 6 | syl 17 | . 2 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
8 | 1, 7 | eqtrid 2789 | 1 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ∅c0 4237 〈cop 4547 × cxp 5549 dom cdm 5551 ‘cfv 6380 (class class class)co 7213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-xp 5557 df-dm 5561 df-iota 6338 df-fv 6388 df-ov 7216 |
This theorem is referenced by: mpondm0 7446 |
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