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Mirrors > Home > MPE Home > Th. List > mressmrcd | Structured version Visualization version GIF version |
Description: In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mressmrcd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mressmrcd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mressmrcd.3 | ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) |
mressmrcd.4 | ⊢ (𝜑 → 𝑇 ⊆ 𝑆) |
Ref | Expression |
---|---|
mressmrcd | ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mressmrcd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | mressmrcd.2 | . . . 4 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | mressmrcd.3 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | |
4 | 1, 2 | mrcssvd 16106 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑋) |
5 | 1, 2, 3, 4 | mrcssd 16107 | . . 3 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘(𝑁‘𝑇))) |
6 | mressmrcd.4 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑆) | |
7 | 3, 4 | sstrd 3578 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
8 | 6, 7 | sstrd 3578 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
9 | 1, 2, 8 | mrcidmd 16109 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑇)) = (𝑁‘𝑇)) |
10 | 5, 9 | sseqtrd 3604 | . 2 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘𝑇)) |
11 | 1, 2, 6, 7 | mrcssd 16107 | . 2 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ (𝑁‘𝑆)) |
12 | 10, 11 | eqssd 3585 | 1 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 Moorecmre 16065 mrClscmrc 16066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-mre 16069 df-mrc 16070 |
This theorem is referenced by: mrieqvlemd 16112 mrissmrcd 16123 |
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