Theorem iscss 21608

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 21607	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
4	3	eleq2d 2814	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}))
5		id 22	. . . 4 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
6		fvex 6839	. . . 4 ⊢ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ V
7	5, 6	eqeltrdi 2836	. . 3 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 ∈ V)
8		id 22	. . . 4 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
9		2fveq3 6831	. . . 4 ⊢ (𝑠 = 𝑆 → ( ⊥ ‘( ⊥ ‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑆)))
10	8, 9	eqeq12d 2745	. . 3 ⊢ (𝑠 = 𝑆 → (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))
11	7, 10	elab3 3644	. 2 ⊢ (𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
12	4, 11	bitrdi 287	1 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3438  ‘cfv 6486  ocvcocv 21585  ClSubSpccss 21586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7356  df-ocv 21588  df-css 21589

This theorem is referenced by:  cssi  21609  iscss2  21611  obslbs  21655  hlhillcs  41940