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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 5702 | . 2 ⊢ Rel (𝑆 × 𝑆) | |
| 2 | opelxp 5720 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) | |
| 3 | df-ov 7435 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
| 4 | oprssdm.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
| 5 | 3, 4 | eqeltrrid 2845 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘〈𝑥, 𝑦〉) ∈ 𝑆) | 
| 6 | oprssdm.1 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑆 | |
| 7 | ndmfv 6940 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ dom 𝐹 → (𝐹‘〈𝑥, 𝑦〉) = ∅) | |
| 8 | 7 | eleq1d 2825 | . . . . . 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 217 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) | 
| 13 | 1, 12 | relssi 5796 | 1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2107 ⊆ wss 3950 ∅c0 4332 〈cop 4631 × cxp 5682 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-dm 5694 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: dmaddsr 11126 dmmulsr 11127 axaddf 11186 axmulf 11187 | 
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