Theorem iscss 20886

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 20885	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
4	3	eleq2d 2826	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}))
5		id 22	. . . 4 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
6		fvex 6784	. . . 4 ⊢ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ V
7	5, 6	eqeltrdi 2849	. . 3 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 ∈ V)
8		id 22	. . . 4 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
9		2fveq3 6776	. . . 4 ⊢ (𝑠 = 𝑆 → ( ⊥ ‘( ⊥ ‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑆)))
10	8, 9	eqeq12d 2756	. . 3 ⊢ (𝑠 = 𝑆 → (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))
11	7, 10	elab3 3619	. 2 ⊢ (𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
12	4, 11	bitrdi 287	1 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2110  {cab 2717  Vcvv 3431  ‘cfv 6432  ocvcocv 20863  ClSubSpccss 20864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-ov 7274  df-ocv 20866  df-css 20867

This theorem is referenced by:  cssi  20887  iscss2  20889  obslbs  20935  hlhillcs  39972