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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 5566 | . 2 ⊢ Rel (𝑆 × 𝑆) | |
2 | opelxp 5584 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) | |
3 | df-ov 7148 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | oprssdm.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | 3, 4 | eqeltrrid 2915 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆) |
6 | oprssdm.1 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑆 | |
7 | ndmfv 6693 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → (𝐹‘〈𝑥, 𝑦〉) = ∅) | |
8 | 7 | eleq1d 2894 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → ((𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
9 | 6, 8 | mtbiri 328 | . . . . 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 218 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
13 | 1, 12 | relssi 5653 | 1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ∅c0 4288 〈cop 4563 × cxp 5546 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: dmaddsr 10495 dmmulsr 10496 axaddf 10555 axmulf 10556 |
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