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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 16894 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑋) |
5 | 1, 2, 3, 4 | mrcssd 16895 | . . 3 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘(𝑁‘𝑇))) |
6 | mressmrcd.4 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑆) | |
7 | 3, 4 | sstrd 3977 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
8 | 6, 7 | sstrd 3977 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
9 | 1, 2, 8 | mrcidmd 16897 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑇)) = (𝑁‘𝑇)) |
10 | 5, 9 | sseqtrd 4007 | . 2 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘𝑇)) |
11 | 1, 2, 6, 7 | mrcssd 16895 | . 2 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ (𝑁‘𝑆)) |
12 | 10, 11 | eqssd 3984 | 1 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 Moorecmre 16853 mrClscmrc 16854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-mre 16857 df-mrc 16858 |
This theorem is referenced by: mrieqvlemd 16900 mrissmrcd 16911 |
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