MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cssval Structured version   Visualization version   GIF version

Theorem cssval 20351
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 3400 . 2 (𝑊𝑋𝑊 ∈ V)
2 cssval.c . . 3 𝐶 = (ClSubSp‘𝑊)
3 fveq2 6411 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
4 cssval.o . . . . . . . 8 = (ocv‘𝑊)
53, 4syl6eqr 2851 . . . . . . 7 (𝑤 = 𝑊 → (ocv‘𝑤) = )
65fveq1d 6413 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( 𝑠))
75, 6fveq12d 6418 . . . . . 6 (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ‘( 𝑠)))
87eqeq2d 2809 . . . . 5 (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ‘( 𝑠))))
98abbidv 2918 . . . 4 (𝑤 = 𝑊 → {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠𝑠 = ( ‘( 𝑠))})
10 df-css 20333 . . . 4 ClSubSp = (𝑤 ∈ V ↦ {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11 fvex 6424 . . . . . 6 (Base‘𝑊) ∈ V
1211pwex 5050 . . . . 5 𝒫 (Base‘𝑊) ∈ V
13 id 22 . . . . . . 7 (𝑠 = ( ‘( 𝑠)) → 𝑠 = ( ‘( 𝑠)))
14 eqid 2799 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
1514, 4ocvss 20339 . . . . . . . 8 ( ‘( 𝑠)) ⊆ (Base‘𝑊)
16 fvex 6424 . . . . . . . . 9 ( ‘( 𝑠)) ∈ V
1716elpw 4355 . . . . . . . 8 (( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ‘( 𝑠)) ⊆ (Base‘𝑊))
1815, 17mpbir 223 . . . . . . 7 ( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊)
1913, 18syl6eqel 2886 . . . . . 6 (𝑠 = ( ‘( 𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
2019abssi 3873 . . . . 5 {𝑠𝑠 = ( ‘( 𝑠))} ⊆ 𝒫 (Base‘𝑊)
2112, 20ssexi 4998 . . . 4 {𝑠𝑠 = ( ‘( 𝑠))} ∈ V
229, 10, 21fvmpt 6507 . . 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 1653  wcel 2157  {cab 2785  Vcvv 3385  wss 3769  𝒫 cpw 4349  cfv 6101  Basecbs 16184  ocvcocv 20329  ClSubSpccss 20330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-ocv 20332  df-css 20333
This theorem is referenced by:  iscss  20352
  Copyright terms: Public domain W3C validator