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Mirrors > Home > MPE Home > Th. List > mrcssd | Structured version Visualization version GIF version |
Description: Moore closure preserves subset ordering. Deduction form of mrcss 16875. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrcssd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mrcssd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mrcssd.3 | ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
mrcssd.4 | ⊢ (𝜑 → 𝑉 ⊆ 𝑋) |
Ref | Expression |
---|---|
mrcssd | ⊢ (𝜑 → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrcssd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | mrcssd.3 | . 2 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) | |
3 | mrcssd.4 | . 2 ⊢ (𝜑 → 𝑉 ⊆ 𝑋) | |
4 | mrcssd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
5 | 4 | mrcss 16875 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) |
6 | 1, 2, 3, 5 | syl3anc 1363 | 1 ⊢ (𝜑 → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 Moorecmre 16841 mrClscmrc 16842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-mre 16845 df-mrc 16846 |
This theorem is referenced by: mressmrcd 16886 mrieqv2d 16898 mrissmrid 16900 mreexexlem2d 16904 isacs3lem 17764 isacs4lem 17766 acsfiindd 17775 acsmapd 17776 acsmap2d 17777 dprdres 19079 dprdss 19080 dprd2dlem1 19092 dprd2da 19093 dmdprdsplit2lem 19096 |
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