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Theorem iscss 20375
 Description: The predicate "is a closed subspace" (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
iscss (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))

Proof of Theorem iscss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 cssval.o . . . 4 = (ocv‘𝑊)
2 cssval.c . . . 4 𝐶 = (ClSubSp‘𝑊)
31, 2cssval 20374 . . 3 (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})
43eleq2d 2878 . 2 (𝑊𝑋 → (𝑆𝐶𝑆 ∈ {𝑠𝑠 = ( ‘( 𝑠))}))
5 id 22 . . . 4 (𝑆 = ( ‘( 𝑆)) → 𝑆 = ( ‘( 𝑆)))
6 fvex 6662 . . . 4 ( ‘( 𝑆)) ∈ V
75, 6eqeltrdi 2901 . . 3 (𝑆 = ( ‘( 𝑆)) → 𝑆 ∈ V)
8 id 22 . . . 4 (𝑠 = 𝑆𝑠 = 𝑆)
9 2fveq3 6654 . . . 4 (𝑠 = 𝑆 → ( ‘( 𝑠)) = ( ‘( 𝑆)))
108, 9eqeq12d 2817 . . 3 (𝑠 = 𝑆 → (𝑠 = ( ‘( 𝑠)) ↔ 𝑆 = ( ‘( 𝑆))))
117, 10elab3 3625 . 2 (𝑆 ∈ {𝑠𝑠 = ( ‘( 𝑠))} ↔ 𝑆 = ( ‘( 𝑆)))
124, 11syl6bb 290 1 (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2112  {cab 2779  Vcvv 3444  ‘cfv 6328  ocvcocv 20352  ClSubSpccss 20353 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-ocv 20355  df-css 20356 This theorem is referenced by:  cssi  20376  iscss2  20378  obslbs  20422  hlhillcs  39247
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