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Mirrors > Home > MPE Home > Th. List > Mathboxes > aoprssdm | Structured version Visualization version GIF version |
Description: Domain of closure of an operation. In contrast to oprssdm 6815, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aoprssdm.1 | ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆) |
Ref | Expression |
---|---|
aoprssdm | ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5227 | . 2 ⊢ Rel (𝑆 × 𝑆) | |
2 | opelxp 5146 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) ↔ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) | |
3 | df-aov 41198 | . . . . 5 ⊢ ((𝑥𝐹𝑦)) = (𝐹'''〈𝑥, 𝑦〉) | |
4 | aoprssdm.1 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆) | |
5 | 3, 4 | syl5eqelr 2706 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐹'''〈𝑥, 𝑦〉) ∈ 𝑆) |
6 | afvvdm 41221 | . . . 4 ⊢ ((𝐹'''〈𝑥, 𝑦〉) ∈ 𝑆 → 〈𝑥, 𝑦〉 ∈ dom 𝐹) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
8 | 2, 7 | sylbi 207 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑆 × 𝑆) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
9 | 1, 8 | relssi 5211 | 1 ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ⊆ wss 3574 〈cop 4183 × cxp 5112 dom cdm 5114 '''cafv 41194 ((caov 41195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 df-rel 5121 df-fv 5896 df-dfat 41196 df-afv 41197 df-aov 41198 |
This theorem is referenced by: (None) |
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