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Theorem cssi 21610
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 6929 . . . 4 (𝑆 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ dom ClSubSp)
2 cssval.c . . . 4 𝐶 = (ClSubSp‘𝑊)
31, 2eleq2s 2847 . . 3 (𝑆𝐶𝑊 ∈ dom ClSubSp)
4 cssval.o . . . 4 = (ocv‘𝑊)
54, 2iscss 21609 . . 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 205   = wceq 1534  wcel 2099  dom cdm 5673  cfv 6543  ocvcocv 21586  ClSubSpccss 21587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-ocv 21589  df-css 21590
This theorem is referenced by:  cssss  21611  cssincl  21614  csslss  21617  cssmre  21619  mrccss  21620  ocvpj  21645  csscld  25171
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