Theorem msrval 31713

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 31712	. . 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 6352	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st ‘𝑠)) ∈ V)
7		fvexd 6352	. . . 4 ⊢ (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) → (2nd ‘𝑠) ∈ V)
8		simpllr 817	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
9	8	fveq2d 6344	. . . . . . . 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 6344	. . . . . . 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 2748	. . . . . . . . . . . . 13 ⊢ (mDV‘𝑇) = (mDV‘𝑇)
12		eqid 2748	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
13	11, 12, 2	elmpst 31711	. . . . . . . . . . . 12 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
14	13	simp1bi 1137	. . . . . . . . . . 11 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ ◡𝐷 = 𝐷))
15	14	simpld 477	. . . . . . . . . 10 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐷 ⊆ (mDV‘𝑇))
16	15	ad3antrrr 768	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝐷 ⊆ (mDV‘𝑇))
17		fvex 6350	. . . . . . . . . 10 ⊢ (mDV‘𝑇) ∈ V
18	17	ssex 4942	. . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
21	20	simprd 482	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐻 ∈ Fin)
22	21	ad3antrrr 768	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝐻 ∈ Fin)
23	13	simp3bi 1139	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐴 ∈ (mEx‘𝑇))
24	23	ad3antrrr 768	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝐴 ∈ (mEx‘𝑇))
25		ot1stg 7335	. . . . . . . 8 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
26	19, 22, 24, 25	syl3anc 1463	. . . . . . 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 2782	. . . . . 6 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (1st ‘(1st ‘𝑠)) = 𝐷)
28		fvex 6350	. . . . . . . . . . 11 ⊢ (mVars‘𝑇) ∈ V
29	1, 28	eqeltri 2823	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
30		imaexg 7256	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → (𝑉 “ (ℎ ∪ {𝑎})) ∈ V)
31	29, 30	ax-mp 5	. . . . . . . . 9 ⊢ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V
32	31	uniex 7106	. . . . . . . 8 ⊢ ∪ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V
33	32	a1i 11	. . . . . . 7 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V)
34		id 22	. . . . . . . . 9 ⊢ (𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎})) → 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
35		simplr 809	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ℎ = (2nd ‘(1st ‘𝑠)))
36	9	fveq2d 6344	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘(1st ‘𝑠)) = (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
37		ot2ndg 7336	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
38	19, 22, 24, 37	syl3anc 1463	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
39	35, 36, 38	3eqtrd 2786	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ℎ = 𝐻)
40		simpr 479	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑎 = (2nd ‘𝑠))
41	8	fveq2d 6344	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
42		ot3rdg 7337	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (mEx‘𝑇) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
43	24, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
44	40, 41, 43	3eqtrd 2786	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → 𝑎 = 𝐴)
45	44	sneqd 4321	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → {𝑎} = {𝐴})
46	39, 45	uneq12d 3899	. . . . . . . . . . . 12 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (ℎ ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
47	46	imaeq2d 5612	. . . . . . . . . . 11 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → (𝑉 “ (ℎ ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
48	47	unieqd 4586	. . . . . . . . . 10 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (𝐻 ∪ {𝐴})))
49		msrval.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
50	48, 49	syl6eqr 2800	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) = 𝑍)
51	34, 50	sylan9eqr 2804	. . . . . . . 8 ⊢ (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) ∧ 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎}))) → 𝑧 = 𝑍)
52	51	sqxpeqd 5286	. . . . . . 7 ⊢ (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) ∧ 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
53	33, 52	csbied 3689	. . . . . 6 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2nd ‘(1st ‘𝑠))) ∧ 𝑎 = (2nd ‘𝑠)) → ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = (𝑍 × 𝑍))
54	27, 53	ineq12d 3946	. . . . 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)))
55	54, 39, 44	oteq123d 4556	. . . 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⟩)
56	7, 55	csbied 3689	. . 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⟩)
57	6, 56	csbied 3689	. 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⟩)
58		id 22	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
59		otex 5070	. . 3 ⊢ ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V
60	59	a1i 11	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V)
61	5, 57, 58, 60	fvmptd 6438	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1620   ∈ wcel 2127  Vcvv 3328  ⦋csb 3662   ∪ cun 3701   ∩ cin 3702   ⊆ wss 3703  {csn 4309  ⟨cotp 4317  ∪ cuni 4576   ↦ cmpt 4869   × cxp 5252  ^◡ccnv 5253   “ cima 5257  ‘cfv 6037  1^st c1st 7319  2^nd c2nd 7320  Fincfn 8109  mExcmex 31642  mDVcmdv 31643  mVarscmvrs 31644  mPreStcmpst 31648  mStRedcmsr 31649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-ot 4318  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-1st 7321  df-2nd 7322  df-mpst 31668  df-msr 31669

This theorem is referenced by:  msrf  31717  msrid  31720  elmsta  31723  mthmpps  31757