Theorem msrval 34132

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 34131	. . 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 6857	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st ‘𝑠)) ∈ V)
7		fvexd 6857	. . . 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 6846	. . . . . . . 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 6846	. . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (mDV‘𝑇) = (mDV‘𝑇)
12		eqid 2736	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
13	11, 12, 2	elmpst 34130	. . . . . . . . . . . 12 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
14	13	simp1bi 1145	. . . . . . . . . . 11 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ ◡𝐷 = 𝐷))
15	14	simpld 495	. . . . . . . . . 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 6855	. . . . . . . . . 10 ⊢ (mDV‘𝑇) ∈ V
18	17	ssex 5278	. . . . . . . . 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 1146	. . . . . . . . . 10 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
21	20	simprd 496	. . . . . . . . 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 1147	. . . . . . . . 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 7935	. . . . . . . 8 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
26	19, 22, 24, 25	syl3anc 1371	. . . . . . 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 2776	. . . . . 6 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (1st ‘(1st ‘𝑠)) = 𝐷)
28	1	fvexi 6856	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
29		imaexg 7852	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → (𝑉 “ (ℎ ∪ {𝑎})) ∈ V)
30	28, 29	ax-mp 5	. . . . . . . . 9 ⊢ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V
31	30	uniex 7678	. . . . . . . 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 6846	. . . . . . . . . . . . . 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 7936	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
37	19, 22, 24, 36	syl3anc 1371	. . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ℎ = 𝐻)
39		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑎 = (2nd ‘𝑠))
40	8	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
41		ot3rdg 7937	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑎 = 𝐴)
44	43	sneqd 4598	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → {𝑎} = {𝐴})
45	38, 44	uneq12d 4124	. . . . . . . . . . . 12 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (ℎ ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
46	45	imaeq2d 6013	. . . . . . . . . . 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 4879	. . . . . . . . . 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	eqtr4di 2794	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) = 𝑍)
50	33, 49	sylan9eqr 2798	. . . . . . . 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 5665	. . . . . . 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 3893	. . . . . 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 4173	. . . . 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 4845	. . . 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 3893	. . 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 3893	. 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 5422	. . 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 6955	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445  ⦋csb 3855   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  {csn 4586  ⟨cotp 4594  ∪ cuni 4865   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632   “ cima 5636  ‘cfv 6496  1^st c1st 7919  2^nd c2nd 7920  Fincfn 8883  mExcmex 34061  mDVcmdv 34062  mVarscmvrs 34063  mPreStcmpst 34067  mStRedcmsr 34068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1st 7921  df-2nd 7922  df-mpst 34087  df-msr 34088

This theorem is referenced by:  msrf  34136  msrid  34139  elmsta  34142  mthmpps  34176