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Theorem cssval 21718
Description: The set of closed subspaces 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
cssval (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
Distinct variable groups:   ,𝑠   𝑊,𝑠
Allowed substitution hints:   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem cssval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3504 . 2 (𝑊𝑋𝑊 ∈ V)
2 cssval.c . . 3 𝐶 = (ClSubSp‘𝑊)
3 fveq2 6919 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
4 cssval.o . . . . . . . 8 = (ocv‘𝑊)
53, 4eqtr4di 2792 . . . . . . 7 (𝑤 = 𝑊 → (ocv‘𝑤) = )
65fveq1d 6921 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( 𝑠))
75, 6fveq12d 6926 . . . . . 6 (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ‘( 𝑠)))
87eqeq2d 2745 . . . . 5 (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ‘( 𝑠))))
98abbidv 2805 . . . 4 (𝑤 = 𝑊 → {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠𝑠 = ( ‘( 𝑠))})
10 df-css 21700 . . . 4 ClSubSp = (𝑤 ∈ V ↦ {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11 fvex 6932 . . . . . 6 (Base‘𝑊) ∈ V
1211pwex 5401 . . . . 5 𝒫 (Base‘𝑊) ∈ V
13 id 22 . . . . . . 7 (𝑠 = ( ‘( 𝑠)) → 𝑠 = ( ‘( 𝑠)))
14 eqid 2734 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
1514, 4ocvss 21706 . . . . . . . 8 ( ‘( 𝑠)) ⊆ (Base‘𝑊)
16 fvex 6932 . . . . . . . . 9 ( ‘( 𝑠)) ∈ V
1716elpw 4626 . . . . . . . 8 (( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ‘( 𝑠)) ⊆ (Base‘𝑊))
1815, 17mpbir 231 . . . . . . 7 ( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊)
1913, 18eqeltrdi 2846 . . . . . 6 (𝑠 = ( ‘( 𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
2019abssi 4087 . . . . 5 {𝑠𝑠 = ( ‘( 𝑠))} ⊆ 𝒫 (Base‘𝑊)
2112, 20ssexi 5343 . . . 4 {𝑠𝑠 = ( ‘( 𝑠))} ∈ V
229, 10, 21fvmpt 7027 . . 3 (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠𝑠 = ( ‘( 𝑠))})
232, 22eqtrid 2786 . 2 (𝑊 ∈ V → 𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
241, 23syl 17 1 (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  {cab 2711  Vcvv 3482  wss 3970  𝒫 cpw 4622  cfv 6572  Basecbs 17253  ocvcocv 21696  ClSubSpccss 21697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580  df-ov 7448  df-ocv 21699  df-css 21700
This theorem is referenced by:  iscss  21719
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