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Theorem cssi 20377
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 6681 . . . 4 (𝑆 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ dom ClSubSp)
2 cssval.c . . . 4 𝐶 = (ClSubSp‘𝑊)
31, 2eleq2s 2911 . . 3 (𝑆𝐶𝑊 ∈ dom ClSubSp)
4 cssval.o . . . 4 = (ocv‘𝑊)
54, 2iscss 20376 . . 3 (𝑊 ∈ dom ClSubSp → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
63, 5syl 17 . 2 (𝑆𝐶 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
76ibi 270 1 (𝑆𝐶𝑆 = ( ‘( 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  dom cdm 5523  cfv 6328  ocvcocv 20353  ClSubSpccss 20354
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-ocv 20356  df-css 20357
This theorem is referenced by:  cssss  20378  cssincl  20381  csslss  20384  cssmre  20386  mrccss  20387  ocvpj  20410  csscld  23857
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