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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 6653 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | ssel2 3598 | . . . . 5 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
3 | opelxp 5146 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
4 | 2, 3 | sylib 208 | . . . 4 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) |
5 | 4 | stoic1a 1697 | . . 3 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
6 | ndmfv 6218 | . . 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 | 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 ⊆ wss 3574 ∅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: mpt2ndm0 6875 |
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