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Theorem iscss 19846
 Description: The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
cssval.o = (ocv‘𝑊)
cssval.c 𝐶 = (CSubSp‘𝑊)
Assertion
Ref Expression
iscss (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))

Proof of Theorem iscss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 cssval.o . . . 4 = (ocv‘𝑊)
2 cssval.c . . . 4 𝐶 = (CSubSp‘𝑊)
31, 2cssval 19845 . . 3 (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
43eleq2d 2673 . 2 (𝑊𝑋 → (𝑆𝐶𝑆 ∈ {𝑠𝑠 = ( ‘( 𝑠))}))
5 id 22 . . . 4 (𝑆 = ( ‘( 𝑆)) → 𝑆 = ( ‘( 𝑆)))
6 fvex 6113 . . . 4 ( ‘( 𝑆)) ∈ V
75, 6syl6eqel 2696 . . 3 (𝑆 = ( ‘( 𝑆)) → 𝑆 ∈ V)
8 id 22 . . . 4 (𝑠 = 𝑆𝑠 = 𝑆)
9 fveq2 6103 . . . . 5 (𝑠 = 𝑆 → ( 𝑠) = ( 𝑆))
109fveq2d 6107 . . . 4 (𝑠 = 𝑆 → ( ‘( 𝑠)) = ( ‘( 𝑆)))
118, 10eqeq12d 2625 . . 3 (𝑠 = 𝑆 → (𝑠 = ( ‘( 𝑠)) ↔ 𝑆 = ( ‘( 𝑆))))
127, 11elab3 3327 . 2 (𝑆 ∈ {𝑠𝑠 = ( ‘( 𝑠))} ↔ 𝑆 = ( ‘( 𝑆)))
134, 12syl6bb 275 1 (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  ‘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:  cssi  19847  iscss2  19849  obslbs  19893  hlhillcs  36268
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