Theorem cssval 21989

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 3471	. 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 2813	. . . . . . 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 2771	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))))
9	8	abbidv 2826	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
10		df-css 21971	. . . 4 ⊢ ClSubSp = (𝑤 ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))})
11		fvex 6898	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
12	11	pwex 5341	. . . . 5 ⊢ 𝒫 (Base‘𝑊) ∈ V
13		id 23	. . . . . . 7 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)))
14		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
15	14, 4	ocvss 21977	. . . . . . . 8 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊)
16		fvex 6898	. . . . . . . . 9 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ V
17	16	elpw 4560	. . . . . . . 8 ⊢ (( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊))
18	15, 17	mpbir 234	. . . . . . 7 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊)
19	13, 18	eqeltrdi 2868	. . . . . 6 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊))
20	19	abssi 3993	. . . . 5 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ⊆ 𝒫 (Base‘𝑊)
21	12, 20	ssexi 5283	. . . 4 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ∈ V
22	9, 10, 21	fvmpt 6993	. . 3 ⊢ (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
23	2, 22	eqtrid 2807	. 2 ⊢ (𝑊 ∈ V → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
24	1, 23	syl 18	1 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3895  𝒫 cpw 4556  ‘cfv 6538  Basecbs 17388  ocvcocv 21967  ClSubSpccss 21968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-ocv 21970  df-css 21971

This theorem is used by:  iscss  21990