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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 6873 | . 2 ⊢ (𝐹‘𝑈) ⊆ ∪ ran 𝐹 | |
2 | mrcfval.f | . . . . 5 ⊢ 𝐹 = (mrCls‘𝐶) | |
3 | 2 | mrcf 17486 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶) |
4 | frn 6673 | . . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
5 | uniss 4872 | . . . 4 ⊢ (ran 𝐹 ⊆ 𝐶 → ∪ ran 𝐹 ⊆ ∪ 𝐶) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ ∪ 𝐶) |
7 | mreuni 17477 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋) | |
8 | 6, 7 | sseqtrd 3983 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ 𝑋) |
9 | 1, 8 | sstrid 3954 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3909 𝒫 cpw 4559 ∪ cuni 4864 ran crn 5633 ⟶wf 6490 ‘cfv 6494 Moorecmre 17459 mrClscmrc 17460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-mre 17463 df-mrc 17464 |
This theorem is referenced by: mrcidb 17492 mrcuni 17498 mrcssvd 17500 mrefg2 41006 proot1hash 41503 |
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