Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prsss | Structured version Visualization version GIF version |
Description: Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
Ref | Expression |
---|---|
ordtNEW.b | ⊢ 𝐵 = (Base‘𝐾) |
ordtNEW.l | ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) |
Ref | Expression |
---|---|
prsss | ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtNEW.l | . . . . 5 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) | |
2 | 1 | ineq1i 4142 | . . . 4 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = (((le‘𝐾) ∩ (𝐵 × 𝐵)) ∩ (𝐴 × 𝐴)) |
3 | inass 4153 | . . . 4 ⊢ (((le‘𝐾) ∩ (𝐵 × 𝐵)) ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) | |
4 | 2, 3 | eqtri 2766 | . . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) |
5 | xpss12 5604 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) ⊆ (𝐵 × 𝐵)) | |
6 | 5 | anidms 567 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐵 × 𝐵)) |
7 | sseqin2 4149 | . . . . 5 ⊢ ((𝐴 × 𝐴) ⊆ (𝐵 × 𝐵) ↔ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) | |
8 | 6, 7 | sylib 217 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
9 | 8 | ineq2d 4146 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((le‘𝐾) ∩ ((𝐵 × 𝐵) ∩ (𝐴 × 𝐴))) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
10 | 4, 9 | eqtrid 2790 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
11 | 10 | adantl 482 | 1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 × cxp 5587 ‘cfv 6433 Basecbs 16912 lecple 16969 Proset cproset 18011 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-in 3894 df-ss 3904 df-opab 5137 df-xp 5595 |
This theorem is referenced by: prsssdm 31867 ordtrestNEW 31871 ordtrest2NEW 31873 |
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