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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 16890. (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 16890 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) |
6 | 1, 2, 3, 5 | syl3anc 1367 | 1 ⊢ (𝜑 → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 Moorecmre 16856 mrClscmrc 16857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-mre 16860 df-mrc 16861 |
This theorem is referenced by: mressmrcd 16901 mrieqv2d 16913 mrissmrid 16915 mreexexlem2d 16919 isacs3lem 17779 isacs4lem 17781 acsfiindd 17790 acsmapd 17791 acsmap2d 17792 dprdres 19153 dprdss 19154 dprd2dlem1 19166 dprd2da 19167 dmdprdsplit2lem 19170 |
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