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Theorem iscss 21103
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 = (ClSubSp‘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 = (ClSubSp‘𝑊)
31, 2cssval 21102 . . 3 (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
43eleq2d 2820 . 2 (𝑊𝑋 → (𝑆𝐶𝑆 ∈ {𝑠𝑠 = ( ‘( 𝑠))}))
5 id 22 . . . 4 (𝑆 = ( ‘( 𝑆)) → 𝑆 = ( ‘( 𝑆)))
6 fvex 6856 . . . 4 ( ‘( 𝑆)) ∈ V
75, 6eqeltrdi 2842 . . 3 (𝑆 = ( ‘( 𝑆)) → 𝑆 ∈ V)
8 id 22 . . . 4 (𝑠 = 𝑆𝑠 = 𝑆)
9 2fveq3 6848 . . . 4 (𝑠 = 𝑆 → ( ‘( 𝑠)) = ( ‘( 𝑆)))
108, 9eqeq12d 2749 . . 3 (𝑠 = 𝑆 → (𝑠 = ( ‘( 𝑠)) ↔ 𝑆 = ( ‘( 𝑆))))
117, 10elab3 3639 . 2 (𝑆 ∈ {𝑠𝑠 = ( ‘( 𝑠))} ↔ 𝑆 = ( ‘( 𝑆)))
124, 11bitrdi 287 1 (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3444  cfv 6497  ocvcocv 21080  ClSubSpccss 21081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-ocv 21083  df-css 21084
This theorem is referenced by:  cssi  21104  iscss2  21106  obslbs  21152  hlhillcs  40471
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