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

Theorem cssval 20496
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 3450 . 2 (𝑊𝑋𝑊 ∈ V)
2 cssval.c . . 3 𝐶 = (ClSubSp‘𝑊)
3 fveq2 6530 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
4 cssval.o . . . . . . . 8 = (ocv‘𝑊)
53, 4syl6eqr 2847 . . . . . . 7 (𝑤 = 𝑊 → (ocv‘𝑤) = )
65fveq1d 6532 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( 𝑠))
75, 6fveq12d 6537 . . . . . 6 (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ‘( 𝑠)))
87eqeq2d 2803 . . . . 5 (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ‘( 𝑠))))
98abbidv 2858 . . . 4 (𝑤 = 𝑊 → {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠𝑠 = ( ‘( 𝑠))})
10 df-css 20478 . . . 4 ClSubSp = (𝑤 ∈ V ↦ {𝑠𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11 fvex 6543 . . . . . 6 (Base‘𝑊) ∈ V
1211pwex 5165 . . . . 5 𝒫 (Base‘𝑊) ∈ V
13 id 22 . . . . . . 7 (𝑠 = ( ‘( 𝑠)) → 𝑠 = ( ‘( 𝑠)))
14 eqid 2793 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
1514, 4ocvss 20484 . . . . . . . 8 ( ‘( 𝑠)) ⊆ (Base‘𝑊)
16 fvex 6543 . . . . . . . . 9 ( ‘( 𝑠)) ∈ V
1716elpw 4453 . . . . . . . 8 (( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ‘( 𝑠)) ⊆ (Base‘𝑊))
1815, 17mpbir 232 . . . . . . 7 ( ‘( 𝑠)) ∈ 𝒫 (Base‘𝑊)
1913, 18syl6eqel 2889 . . . . . 6 (𝑠 = ( ‘( 𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
2019abssi 3962 . . . . 5 {𝑠𝑠 = ( ‘( 𝑠))} ⊆ 𝒫 (Base‘𝑊)
2112, 20ssexi 5110 . . . 4 {𝑠𝑠 = ( ‘( 𝑠))} ∈ V
229, 10, 21fvmpt 6626 . . 3 (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠𝑠 = ( ‘( 𝑠))})
232, 22syl5eq 2841 . 2 (𝑊 ∈ V → 𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
241, 23syl 17 1 (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1520  wcel 2079  {cab 2773  Vcvv 3432  wss 3854  𝒫 cpw 4447  cfv 6217  Basecbs 16300  ocvcocv 20474  ClSubSpccss 20475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fv 6225  df-ov 7010  df-ocv 20477  df-css 20478
This theorem is referenced by:  iscss  20497
  Copyright terms: Public domain W3C validator