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Mirrors > Home > MPE Home > Th. List > mrcssv | Structured version Visualization version GIF version |
Description: The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mrcfval.f | ⊢ 𝐹 = (mrCls‘𝐶) |
Ref | Expression |
---|---|
mrcssv | ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvssunirn 6692 | . 2 ⊢ (𝐹‘𝑈) ⊆ ∪ ran 𝐹 | |
2 | mrcfval.f | . . . . 5 ⊢ 𝐹 = (mrCls‘𝐶) | |
3 | 2 | mrcf 16868 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶) |
4 | frn 6513 | . . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
5 | uniss 4851 | . . . 4 ⊢ (ran 𝐹 ⊆ 𝐶 → ∪ ran 𝐹 ⊆ ∪ 𝐶) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ ∪ 𝐶) |
7 | mreuni 16859 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋) | |
8 | 6, 7 | sseqtrd 4004 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ 𝑋) |
9 | 1, 8 | sstrid 3975 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ran crn 5549 ⟶wf 6344 ‘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-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: mrcidb 16874 mrcuni 16880 mrcssvd 16882 mrefg2 39182 proot1hash 39678 |
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