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Mirrors > Home > MPE Home > Th. List > oprssdm | Structured version Visualization version GIF version |
Description: Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
oprssdm.1 | ⊢ ¬ ∅ ∈ 𝑆 |
oprssdm.2 | ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
oprssdm | ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5150 | . 2 ⊢ Rel (𝑆 × 𝑆) | |
2 | opelxp 5070 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) | |
3 | df-ov 6552 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | oprssdm.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | 3, 4 | syl5eqelr 2693 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆) |
6 | oprssdm.1 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑆 | |
7 | ndmfv 6128 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → (𝐹‘〈𝑥, 𝑦〉) = ∅) | |
8 | 7 | eleq1d 2672 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → ((𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
9 | 6, 8 | mtbiri 316 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → ¬ (𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆) |
10 | 9 | con4i 112 | . . . 4 ⊢ ((𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆 → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
11 | 5, 10 | syl 17 | . . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
12 | 2, 11 | sylbi 206 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
13 | 1, 12 | relssi 5134 | 1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 〈cop 4131 × cxp 5036 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 |
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-8 1979 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-pow 4769 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-eu 2462 df-mo 2463 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-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: dmaddsr 9785 dmmulsr 9786 axaddf 9845 axmulf 9846 |
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