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Theorem cssval 20375
 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 3459 . 2 (𝑊𝑋𝑊 ∈ V)
2 cssval.c . . 3 𝐶 = (ClSubSp‘𝑊)
3 fveq2 6645 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
4 cssval.o . . . . . . . 8 = (ocv‘𝑊)
53, 4eqtr4di 2851 . . . . . . 7 (𝑤 = 𝑊 → (ocv‘𝑤) = )
65fveq1d 6647 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( 𝑠))
75, 6fveq12d 6652 . . . . . 6 (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ‘( 𝑠)))
87eqeq2d 2809 . . . . 5 (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ‘( 𝑠))))
98abbidv 2862 . . . 4 (𝑤 = 𝑊 → {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠𝑠 = ( ‘( 𝑠))})
10 df-css 20357 . . . 4 ClSubSp = (𝑤 ∈ V ↦ {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11 fvex 6658 . . . . . 6 (Base‘𝑊) ∈ V
1211pwex 5246 . . . . 5 𝒫 (Base‘𝑊) ∈ V
13 id 22 . . . . . . 7 (𝑠 = ( ‘( 𝑠)) → 𝑠 = ( ‘( 𝑠)))
14 eqid 2798 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
1514, 4ocvss 20363 . . . . . . . 8 ( ‘( 𝑠)) ⊆ (Base‘𝑊)
16 fvex 6658 . . . . . . . . 9 ( ‘( 𝑠)) ∈ V
1716elpw 4501 . . . . . . . 8 (( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ‘( 𝑠)) ⊆ (Base‘𝑊))
1815, 17mpbir 234 . . . . . . 7 ( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊)
1913, 18eqeltrdi 2898 . . . . . 6 (𝑠 = ( ‘( 𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
2019abssi 3997 . . . . 5 {𝑠𝑠 = ( ‘( 𝑠))} ⊆ 𝒫 (Base‘𝑊)
2112, 20ssexi 5190 . . . 4 {𝑠𝑠 = ( ‘( 𝑠))} ∈ V
229, 10, 21fvmpt 6745 . . 3 (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠𝑠 = ( ‘( 𝑠))})
232, 22syl5eq 2845 . 2 (𝑊 ∈ V → 𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
241, 23syl 17 1 (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  ‘cfv 6324  Basecbs 16477  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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-ocv 20356  df-css 20357 This theorem is referenced by:  iscss  20376
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