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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 5603 | . 2 ⊢ Rel (𝑆 × 𝑆) | |
2 | opelxp 5621 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) | |
3 | df-ov 7271 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | oprssdm.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | 3, 4 | eqeltrrid 2844 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆) |
6 | oprssdm.1 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑆 | |
7 | ndmfv 6797 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → (𝐹‘〈𝑥, 𝑦〉) = ∅) | |
8 | 7 | eleq1d 2823 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → ((𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
9 | 6, 8 | mtbiri 327 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → ¬ (𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆) |
10 | 9 | con4i 114 | . . . 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 216 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
13 | 1, 12 | relssi 5691 | 1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3887 ∅c0 4257 〈cop 4568 × cxp 5583 dom cdm 5585 ‘cfv 6427 (class class class)co 7268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-xp 5591 df-rel 5592 df-dm 5595 df-iota 6385 df-fv 6435 df-ov 7271 |
This theorem is referenced by: dmaddsr 10829 dmmulsr 10830 axaddf 10889 axmulf 10890 |
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