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Theorem cssi 19847
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 = (CSubSp‘𝑊)
Assertion
Ref Expression
cssi (𝑆𝐶𝑆 = ( ‘( 𝑆)))

Proof of Theorem cssi
StepHypRef Expression
1 elfvdm 6130 . . . 4 (𝑆 ∈ (CSubSp‘𝑊) → 𝑊 ∈ dom CSubSp)
2 cssval.c . . . 4 𝐶 = (CSubSp‘𝑊)
31, 2eleq2s 2706 . . 3 (𝑆𝐶𝑊 ∈ dom CSubSp)
4 cssval.o . . . 4 = (ocv‘𝑊)
54, 2iscss 19846 . . 3 (𝑊 ∈ dom CSubSp → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
63, 5syl 17 . 2 (𝑆𝐶 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
76ibi 255 1 (𝑆𝐶𝑆 = ( ‘( 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wcel 1977  dom cdm 5038  cfv 5804  ocvcocv 19823  CSubSpccss 19824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-ocv 19826  df-css 19827
This theorem is referenced by:  cssss  19848  cssincl  19851  csslss  19854  cssmre  19856  mrccss  19857  ocvpj  19880  csscld  22856
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