Theorem msrval 32787

Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
msrfval.v	⊢ 𝑉 = (mVars‘𝑇)
msrfval.p	⊢ 𝑃 = (mPreSt‘𝑇)
msrfval.r	⊢ 𝑅 = (mStRed‘𝑇)
msrval.z	⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))

Assertion

Ref	Expression
msrval	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)

Proof of Theorem msrval

Dummy variables ℎ 𝑎 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		msrfval.v	. . . 4 ⊢ 𝑉 = (mVars‘𝑇)
2		msrfval.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
3		msrfval.r	. . . 4 ⊢ 𝑅 = (mStRed‘𝑇)
4	1, 2, 3	msrfval 32786	. . 3 ⊢ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
5	4	a1i 11	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
6		fvexd 6687	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st ‘𝑠)) ∈ V)
7		fvexd 6687	. . . 4 ⊢ (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) → (2nd ‘𝑠) ∈ V)
8		simpllr 774	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
9	8	fveq2d 6676	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (1st ‘𝑠) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
10	9	fveq2d 6676	. . . . . . 7 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (1st ‘(1st ‘𝑠)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
11		eqid 2823	. . . . . . . . . . . . 13 ⊢ (mDV‘𝑇) = (mDV‘𝑇)
12		eqid 2823	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
13	11, 12, 2	elmpst 32785	. . . . . . . . . . . 12 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
14	13	simp1bi 1141	. . . . . . . . . . 11 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ ◡𝐷 = 𝐷))
15	14	simpld 497	. . . . . . . . . 10 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐷 ⊆ (mDV‘𝑇))
16	15	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝐷 ⊆ (mDV‘𝑇))
17		fvex 6685	. . . . . . . . . 10 ⊢ (mDV‘𝑇) ∈ V
18	17	ssex 5227	. . . . . . . . 9 ⊢ (𝐷 ⊆ (mDV‘𝑇) → 𝐷 ∈ V)
19	16, 18	syl 17	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝐷 ∈ V)
20	13	simp2bi 1142	. . . . . . . . . 10 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
21	20	simprd 498	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐻 ∈ Fin)
22	21	ad3antrrr 728	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝐻 ∈ Fin)
23	13	simp3bi 1143	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐴 ∈ (mEx‘𝑇))
24	23	ad3antrrr 728	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝐴 ∈ (mEx‘𝑇))
25		ot1stg 7705	. . . . . . . 8 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
26	19, 22, 24, 25	syl3anc 1367	. . . . . . 7 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
27	10, 26	eqtrd 2858	. . . . . 6 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (1st ‘(1st ‘𝑠)) = 𝐷)
28	1	fvexi 6686	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
29		imaexg 7622	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → (𝑉 “ (ℎ ∪ {𝑎})) ∈ V)
30	28, 29	ax-mp 5	. . . . . . . . 9 ⊢ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V
31	30	uniex 7469	. . . . . . . 8 ⊢ ∪ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V
32	31	a1i 11	. . . . . . 7 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V)
33		id 22	. . . . . . . . 9 ⊢ (𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎})) → 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
34		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ℎ = (2nd ‘(1st ‘𝑠)))
35	9	fveq2d 6676	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘(1st ‘𝑠)) = (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
36		ot2ndg 7706	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
37	19, 22, 24, 36	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
38	34, 35, 37	3eqtrd 2862	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ℎ = 𝐻)
39		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑎 = (2nd ‘𝑠))
40	8	fveq2d 6676	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
41		ot3rdg 7707	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (mEx‘𝑇) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
42	24, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
43	39, 40, 42	3eqtrd 2862	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑎 = 𝐴)
44	43	sneqd 4581	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → {𝑎} = {𝐴})
45	38, 44	uneq12d 4142	. . . . . . . . . . . 12 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (ℎ ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
46	45	imaeq2d 5931	. . . . . . . . . . 11 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (𝑉 “ (ℎ ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
47	46	unieqd 4854	. . . . . . . . . 10 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (𝐻 ∪ {𝐴})))
48		msrval.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
49	47, 48	syl6eqr 2876	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) = 𝑍)
50	33, 49	sylan9eqr 2880	. . . . . . . 8 ⊢ (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) ∧ 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎}))) → 𝑧 = 𝑍)
51	50	sqxpeqd 5589	. . . . . . 7 ⊢ (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) ∧ 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
52	32, 51	csbied 3921	. . . . . 6 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = (𝑍 × 𝑍))
53	27, 52	ineq12d 4192	. . . . 5 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)) = (𝐷 ∩ (𝑍 × 𝑍)))
54	53, 38, 43	oteq123d 4820	. . . 4 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
55	7, 54	csbied 3921	. . 3 ⊢ (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) → ⦋(2nd ‘𝑠) / 𝑎⦌⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
56	6, 55	csbied 3921	. 2 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
57		id 22	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
58		otex 5359	. . 3 ⊢ ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V
59	58	a1i 11	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V)
60	5, 56, 57, 59	fvmptd 6777	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ⦋csb 3885   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  {csn 4569  ⟨cotp 4577  ∪ cuni 4840   ↦ cmpt 5148   × cxp 5555  ^◡ccnv 5556   “ cima 5560  ‘cfv 6357  1^st c1st 7689  2^nd c2nd 7690  Fincfn 8511  mExcmex 32716  mDVcmdv 32717  mVarscmvrs 32718  mPreStcmpst 32722  mStRedcmsr 32723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-ot 4578  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-1st 7691  df-2nd 7692  df-mpst 32742  df-msr 32743

This theorem is referenced by:  msrf  32791  msrid  32794  elmsta  32797  mthmpps  32831