![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5656 | . 2 ⊢ Rel (𝑆 × 𝑆) | |
2 | opelxp 5674 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) | |
3 | df-ov 7365 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩) | |
4 | oprssdm.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | 3, 4 | eqeltrrid 2843 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆) |
6 | oprssdm.1 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑆 | |
7 | ndmfv 6882 | . . . . . . 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 5748 | 1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3915 ∅c0 4287 ⟨cop 4597 × cxp 5636 dom cdm 5638 ‘cfv 6501 (class class class)co 7362 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-dm 5648 df-iota 6453 df-fv 6509 df-ov 7365 |
This theorem is referenced by: dmaddsr 11028 dmmulsr 11029 axaddf 11088 axmulf 11089 |
Copyright terms: Public domain | W3C validator |