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Theorem cssval 21631
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 3480 . 2 (𝑊𝑋𝑊 ∈ V)
2 cssval.c . . 3 𝐶 = (ClSubSp‘𝑊)
3 fveq2 6896 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
4 cssval.o . . . . . . . 8 = (ocv‘𝑊)
53, 4eqtr4di 2783 . . . . . . 7 (𝑤 = 𝑊 → (ocv‘𝑤) = )
65fveq1d 6898 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( 𝑠))
75, 6fveq12d 6903 . . . . . 6 (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ‘( 𝑠)))
87eqeq2d 2736 . . . . 5 (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ‘( 𝑠))))
98abbidv 2794 . . . 4 (𝑤 = 𝑊 → {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠𝑠 = ( ‘( 𝑠))})
10 df-css 21613 . . . 4 ClSubSp = (𝑤 ∈ V ↦ {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11 fvex 6909 . . . . . 6 (Base‘𝑊) ∈ V
1211pwex 5380 . . . . 5 𝒫 (Base‘𝑊) ∈ V
13 id 22 . . . . . . 7 (𝑠 = ( ‘( 𝑠)) → 𝑠 = ( ‘( 𝑠)))
14 eqid 2725 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
1514, 4ocvss 21619 . . . . . . . 8 ( ‘( 𝑠)) ⊆ (Base‘𝑊)
16 fvex 6909 . . . . . . . . 9 ( ‘( 𝑠)) ∈ V
1716elpw 4608 . . . . . . . 8 (( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ‘( 𝑠)) ⊆ (Base‘𝑊))
1815, 17mpbir 230 . . . . . . 7 ( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊)
1913, 18eqeltrdi 2833 . . . . . 6 (𝑠 = ( ‘( 𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
2019abssi 4063 . . . . 5 {𝑠𝑠 = ( ‘( 𝑠))} ⊆ 𝒫 (Base‘𝑊)
2112, 20ssexi 5323 . . . 4 {𝑠𝑠 = ( ‘( 𝑠))} ∈ V
229, 10, 21fvmpt 7004 . . 3 (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠𝑠 = ( ‘( 𝑠))})
232, 22eqtrid 2777 . 2 (𝑊 ∈ V → 𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
241, 23syl 17 1 (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {cab 2702  Vcvv 3461  wss 3944  𝒫 cpw 4604  cfv 6549  Basecbs 17183  ocvcocv 21609  ClSubSpccss 21610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-ocv 21612  df-css 21613
This theorem is referenced by:  iscss  21632
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