msrval - Mathbox for Mario Carneiro

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Theorem msrval 34560

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 34559	. . 3 ⊢ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
5	4	a1i 11	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
6		fvexd 6907	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2^nd ‘(1^st ‘𝑠)) ∈ V)
7		fvexd 6907	. . . 4 ⊢ (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) → (2^nd ‘𝑠) ∈ V)
8		simpllr 775	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
9	8	fveq2d 6896	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (1^st ‘𝑠) = (1^st ‘⟨𝐷, 𝐻, 𝐴⟩))
10	9	fveq2d 6896	. . . . . . 7 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (1^st ‘(1^st ‘𝑠)) = (1^st ‘(1^st ‘⟨𝐷, 𝐻, 𝐴⟩)))
11		eqid 2733	. . . . . . . . . . . . 13 ⊢ (mDV‘𝑇) = (mDV‘𝑇)
12		eqid 2733	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
13	11, 12, 2	elmpst 34558	. . . . . . . . . . . 12 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ ^◡𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
14	13	simp1bi 1146	. . . . . . . . . . 11 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ ^◡𝐷 = 𝐷))
15	14	simpld 496	. . . . . . . . . 10 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐷 ⊆ (mDV‘𝑇))
16	15	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → 𝐷 ⊆ (mDV‘𝑇))
17		fvex 6905	. . . . . . . . . 10 ⊢ (mDV‘𝑇) ∈ V
18	17	ssex 5322	. . . . . . . . 9 ⊢ (𝐷 ⊆ (mDV‘𝑇) → 𝐷 ∈ V)
19	16, 18	syl 17	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → 𝐷 ∈ V)
20	13	simp2bi 1147	. . . . . . . . . 10 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
21	20	simprd 497	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐻 ∈ Fin)
22	21	ad3antrrr 729	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → 𝐻 ∈ Fin)
23	13	simp3bi 1148	. . . . . . . . 9 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → 𝐴 ∈ (mEx‘𝑇))
24	23	ad3antrrr 729	. . . . . . . 8 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → 𝐴 ∈ (mEx‘𝑇))
25		ot1stg 7989	. . . . . . . 8 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1^st ‘(1^st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
26	19, 22, 24, 25	syl3anc 1372	. . . . . . 7 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (1^st ‘(1^st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
27	10, 26	eqtrd 2773	. . . . . 6 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (1^st ‘(1^st ‘𝑠)) = 𝐷)
28	1	fvexi 6906	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
29		imaexg 7906	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → (𝑉 “ (ℎ ∪ {𝑎})) ∈ V)
30	28, 29	ax-mp 5	. . . . . . . . 9 ⊢ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V
31	30	uniex 7731	. . . . . . . 8 ⊢ ∪ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V
32	31	a1i 11	. . . . . . 7 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) ∈ V)
33		id 22	. . . . . . . . 9 ⊢ (𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎})) → 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
34		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ℎ = (2^nd ‘(1^st ‘𝑠)))
35	9	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (2^nd ‘(1^st ‘𝑠)) = (2^nd ‘(1^st ‘⟨𝐷, 𝐻, 𝐴⟩)))
36		ot2ndg 7990	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2^nd ‘(1^st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
37	19, 22, 24, 36	syl3anc 1372	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (2^nd ‘(1^st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
38	34, 35, 37	3eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ℎ = 𝐻)
39		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → 𝑎 = (2^nd ‘𝑠))
40	8	fveq2d 6896	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (2^nd ‘𝑠) = (2^nd ‘⟨𝐷, 𝐻, 𝐴⟩))
41		ot3rdg 7991	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (mEx‘𝑇) → (2^nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
42	24, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (2^nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
43	39, 40, 42	3eqtrd 2777	. . . . . . . . . . . . . 14 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → 𝑎 = 𝐴)
44	43	sneqd 4641	. . . . . . . . . . . . 13 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → {𝑎} = {𝐴})
45	38, 44	uneq12d 4165	. . . . . . . . . . . 12 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (ℎ ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
46	45	imaeq2d 6060	. . . . . . . . . . 11 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → (𝑉 “ (ℎ ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
47	46	unieqd 4923	. . . . . . . . . 10 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (𝐻 ∪ {𝐴})))
48		msrval.z	. . . . . . . . . 10 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
49	47, 48	eqtr4di 2791	. . . . . . . . 9 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ∪ (𝑉 “ (ℎ ∪ {𝑎})) = 𝑍)
50	33, 49	sylan9eqr 2795	. . . . . . . 8 ⊢ (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) ∧ 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎}))) → 𝑧 = 𝑍)
51	50	sqxpeqd 5709	. . . . . . 7 ⊢ (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) ∧ 𝑧 = ∪ (𝑉 “ (ℎ ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
52	32, 51	csbied 3932	. . . . . 6 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = (𝑍 × 𝑍))
53	27, 52	ineq12d 4214	. . . . 5 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)) = (𝐷 ∩ (𝑍 × 𝑍)))
54	53, 38, 43	oteq123d 4889	. . . 4 ⊢ ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) ∧ 𝑎 = (2^nd ‘𝑠)) → ⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
55	7, 54	csbied 3932	. . 3 ⊢ (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ ℎ = (2^nd ‘(1^st ‘𝑠))) → ⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
56	6, 55	csbied 3932	. 2 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘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 5466	. . 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 7006	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⦋csb 3894 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 {csn 4629 ⟨cotp 4637 ∪ cuni 4909 ↦ cmpt 5232 × cxp 5675 ^◡ccnv 5676 “ cima 5680 ‘cfv 6544 1^st c1st 7973 2^nd c2nd 7974 Fincfn 8939 mExcmex 34489 mDVcmdv 34490 mVarscmvrs 34491 mPreStcmpst 34495 mStRedcmsr 34496

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-1st 7975 df-2nd 7976 df-mpst 34515 df-msr 34516

This theorem is referenced by: msrf 34564 msrid 34567 elmsta 34570 mthmpps 34604

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