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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 5537 | . 2 ⊢ Rel (𝑆 × 𝑆) | |
2 | opelxp 5555 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) | |
3 | df-ov 7138 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | oprssdm.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | 3, 4 | eqeltrrid 2895 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆) |
6 | oprssdm.1 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑆 | |
7 | ndmfv 6675 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → (𝐹‘〈𝑥, 𝑦〉) = ∅) | |
8 | 7 | eleq1d 2874 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → ((𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
9 | 6, 8 | mtbiri 330 | . . . . 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 220 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
13 | 1, 12 | relssi 5624 | 1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 〈cop 4531 × cxp 5517 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: dmaddsr 10496 dmmulsr 10497 axaddf 10556 axmulf 10557 |
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