Theorem iscss 21814

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	⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))

Proof of Theorem iscss

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cssval.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
2		cssval.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊)
3	1, 2	cssval 21813	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
4	3	eleq2d 2849	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}))
5		id 23	. . . 4 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
6		fvex 6896	. . . 4 ⊢ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ V
7	5, 6	eqeltrdi 2871	. . 3 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 ∈ V)
8		id 23	. . . 4 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
9		2fveq3 6888	. . . 4 ⊢ (𝑠 = 𝑆 → ( ⊥ ‘( ⊥ ‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑆)))
10	8, 9	eqeq12d 2779	. . 3 ⊢ (𝑠 = 𝑆 → (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))
11	7, 10	elab3 3646	. 2 ⊢ (𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
12	4, 11	bitrdi 290	1 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {cab 2741  Vcvv 3455  ‘cfv 6538  ocvcocv 21791  ClSubSpccss 21792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-ocv 21794  df-css 21795

This theorem is referenced by:  cssi  21815  iscss2  21817  obslbs  21861  hlhillcs  42713