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Mirrors > Home > MPE Home > Th. List > cssss | Structured version Visualization version GIF version |
Description: A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cssss.v | ⊢ 𝑉 = (Base‘𝑊) |
cssss.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
cssss | ⊢ (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
2 | cssss.c | . . 3 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | 1, 2 | cssi 20822 | . 2 ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑆))) |
4 | cssss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
5 | 4, 1 | ocvss 20808 | . 2 ⊢ ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑆)) ⊆ 𝑉 |
6 | 3, 5 | eqsstrdi 4021 | 1 ⊢ (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ‘cfv 6350 Basecbs 16477 ocvcocv 20798 ClSubSpccss 20799 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-ocv 20801 df-css 20802 |
This theorem is referenced by: cssmre 20831 ocvpj 20855 hlhillcs 39088 |
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