Theorem cssval 20799

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	⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘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.

Step	Hyp	Ref	Expression
1		elex 3440	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		cssval.c	. . 3 ⊢ 𝐶 = (ClSubSp‘𝑊)
3		fveq2 6756	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
4		cssval.o	. . . . . . . 8 ⊢ ⊥ = (ocv‘𝑊)
5	3, 4	eqtr4di 2797	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = ⊥ )
6	5	fveq1d 6758	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( ⊥ ‘𝑠))
7	5, 6	fveq12d 6763	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠)))
8	7	eqeq2d 2749	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))))
9	8	abbidv 2808	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
10		df-css 20781	. . . 4 ⊢ ClSubSp = (𝑤 ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11		fvex 6769	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
12	11	pwex 5298	. . . . 5 ⊢ 𝒫 (Base‘𝑊) ∈ V
13		id 22	. . . . . . 7 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)))
14		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
15	14, 4	ocvss 20787	. . . . . . . 8 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊)
16		fvex 6769	. . . . . . . . 9 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ V
17	16	elpw 4534	. . . . . . . 8 ⊢ (( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊))
18	15, 17	mpbir 230	. . . . . . 7 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊)
19	13, 18	eqeltrdi 2847	. . . . . 6 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
20	19	abssi 3999	. . . . 5 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ⊆ 𝒫 (Base‘𝑊)
21	12, 20	ssexi 5241	. . . 4 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ∈ V
22	9, 10, 21	fvmpt 6857	. . 3 ⊢ (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
23	2, 22	eqtrid 2790	. 2 ⊢ (𝑊 ∈ V → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
24	1, 23	syl 17	1 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {cab 2715  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ‘cfv 6418  Basecbs 16840  ocvcocv 20777  ClSubSpccss 20778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-ocv 20780  df-css 20781

This theorem is referenced by:  iscss  20800