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Theorem cssi 21727
Description: Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
cssval.o = (ocv‘𝑊)
cssval.c 𝐶 = (ClSubSp‘𝑊)
Assertion
Ref Expression
cssi (𝑆𝐶𝑆 = ( ‘( 𝑆)))

Proof of Theorem cssi
StepHypRef Expression
1 elfvdm 6959 . . . 4 (𝑆 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ dom ClSubSp)
2 cssval.c . . . 4 𝐶 = (ClSubSp‘𝑊)
31, 2eleq2s 2862 . . 3 (𝑆𝐶𝑊 ∈ dom ClSubSp)
4 cssval.o . . . 4 = (ocv‘𝑊)
54, 2iscss 21726 . . 3 (𝑊 ∈ dom ClSubSp → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
63, 5syl 17 . 2 (𝑆𝐶 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
76ibi 267 1 (𝑆𝐶𝑆 = ( ‘( 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2108  dom cdm 5700  cfv 6575  ocvcocv 21703  ClSubSpccss 21704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583  df-ov 7453  df-ocv 21706  df-css 21707
This theorem is referenced by:  cssss  21728  cssincl  21731  csslss  21734  cssmre  21736  mrccss  21737  ocvpj  21762  csscld  25304
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