Theorem cssval 21653

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 3484	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		cssval.c	. . 3 ⊢ 𝐶 = (ClSubSp‘𝑊)
3		fveq2 6885	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
4		cssval.o	. . . . . . . 8 ⊢ ⊥ = (ocv‘𝑊)
5	3, 4	eqtr4di 2787	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = ⊥ )
6	5	fveq1d 6887	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( ⊥ ‘𝑠))
7	5, 6	fveq12d 6892	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠)))
8	7	eqeq2d 2745	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))))
9	8	abbidv 2800	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
10		df-css 21635	. . . 4 ⊢ ClSubSp = (𝑤 ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11		fvex 6898	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
12	11	pwex 5360	. . . . 5 ⊢ 𝒫 (Base‘𝑊) ∈ V
13		id 22	. . . . . . 7 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)))
14		eqid 2734	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
15	14, 4	ocvss 21641	. . . . . . . 8 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊)
16		fvex 6898	. . . . . . . . 9 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ V
17	16	elpw 4584	. . . . . . . 8 ⊢ (( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊))
18	15, 17	mpbir 231	. . . . . . 7 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊)
19	13, 18	eqeltrdi 2841	. . . . . 6 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
20	19	abssi 4050	. . . . 5 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ⊆ 𝒫 (Base‘𝑊)
21	12, 20	ssexi 5302	. . . 4 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ∈ V
22	9, 10, 21	fvmpt 6995	. . 3 ⊢ (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
23	2, 22	eqtrid 2781	. 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 2107  {cab 2712  Vcvv 3463   ⊆ wss 3931  𝒫 cpw 4580  ‘cfv 6540  Basecbs 17228  ocvcocv 21631  ClSubSpccss 21632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7415  df-ocv 21634  df-css 21635

This theorem is referenced by:  iscss  21654