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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 17661. (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 17661 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) |
6 | 1, 2, 3, 5 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 Moorecmre 17627 mrClscmrc 17628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-mre 17631 df-mrc 17632 |
This theorem is referenced by: mressmrcd 17672 mrieqv2d 17684 mrissmrid 17686 mreexexlem2d 17690 isacs3lem 18600 isacs4lem 18602 acsfiindd 18611 acsmapd 18612 acsmap2d 18613 dprdres 20063 dprdss 20064 dprd2dlem1 20076 dprd2da 20077 dmdprdsplit2lem 20080 |
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