msubvrs - Mathbox for Mario Carneiro

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Theorem msubvrs 33189

Description: The set of variables in a substitution is the union, indexed by the variables in the original expression, of the variables in the substitution to that variable. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
msubvrs.s	⊢ 𝑆 = (mSubst‘𝑇)
msubvrs.e	⊢ 𝐸 = (mEx‘𝑇)
msubvrs.v	⊢ 𝑉 = (mVars‘𝑇)
msubvrs.h	⊢ 𝐻 = (mVH‘𝑇)

**Assertion**
Ref	Expression
msubvrs	⊢ ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝐸) → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))

Distinct variable groups: 𝑥,𝐸 𝑥,𝐹 𝑥,𝑇 𝑥,𝑋 𝑥,𝑉 Allowed substitution hints: 𝑆(𝑥) 𝐻(𝑥)

Proof of Theorem msubvrs Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		msubvrs.e	. . . . . 6 ⊢ 𝐸 = (mEx‘𝑇)
2		eqid 2736	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubvrs.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 33157	. . . . 5 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
5	4	eleq2i 2822	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
6		eqid 2736	. . . . 5 ⊢ (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
7	1	fvexi 6709	. . . . . 6 ⊢ 𝐸 ∈ V
8	7	mptex 7017	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) ∈ V
9	6, 8	elrnmpti 5814	. . . 4 ⊢ (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
10	5, 9	bitri 278	. . 3 ⊢ (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
11		simp2 1139	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑓 ∈ ran (mRSubst‘𝑇))
12		simp3 1140	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
13		eqid 2736	. . . . . . . . . . . 12 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
14		eqid 2736	. . . . . . . . . . . 12 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
15	13, 1, 14	mexval 33131	. . . . . . . . . . 11 ⊢ 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
16	12, 15	eleqtrdi 2841	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17		xp2nd 7772	. . . . . . . . . 10 ⊢ (𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (2^nd ‘𝑋) ∈ (mREx‘𝑇))
18	16, 17	syl 17	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (2^nd ‘𝑋) ∈ (mREx‘𝑇))
19		eqid 2736	. . . . . . . . . 10 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
20	2, 19, 14	mrsubvrs 33151	. . . . . . . . 9 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ (2^nd ‘𝑋) ∈ (mREx‘𝑇)) → (ran (𝑓‘(2^nd ‘𝑋)) ∩ (mVR‘𝑇)) = ∪ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
21	11, 18, 20	syl2anc 587	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (ran (𝑓‘(2^nd ‘𝑋)) ∩ (mVR‘𝑇)) = ∪ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
22		fveq2 6695	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (1^st ‘𝑒) = (1^st ‘𝑋))
23		2fveq3 6700	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘𝑋)))
24	22, 23	opeq12d 4778	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑋 → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩)
25		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)
26		opex 5333	. . . . . . . . . . . 12 ⊢ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ ∈ V
27	24, 25, 26	fvmpt3i 6801	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐸 → ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋) = ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩)
28	12, 27	syl 17	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋) = ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩)
29	28	fveq2d 6699	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋)) = (𝑉‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩))
30		xp1st 7771	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (1^st ‘𝑋) ∈ (mTC‘𝑇))
31	16, 30	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (1^st ‘𝑋) ∈ (mTC‘𝑇))
32	2, 14	mrsubf 33146	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ran (mRSubst‘𝑇) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
33	11, 32	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
34	17, 15	eleq2s 2849	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐸 → (2^nd ‘𝑋) ∈ (mREx‘𝑇))
35	12, 34	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (2^nd ‘𝑋) ∈ (mREx‘𝑇))
36	33, 35	ffvelrnd 6883	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑓‘(2^nd ‘𝑋)) ∈ (mREx‘𝑇))
37		opelxpi 5573	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑋) ∈ (mTC‘𝑇) ∧ (𝑓‘(2^nd ‘𝑋)) ∈ (mREx‘𝑇)) → ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
38	31, 36, 37	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
39	38, 15	eleqtrrdi 2842	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩ ∈ 𝐸)
40		msubvrs.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVars‘𝑇)
41	19, 1, 40	mvrsval 33134	. . . . . . . . . 10 ⊢ (⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩ ∈ 𝐸 → (𝑉‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) = (ran (2^nd ‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) ∩ (mVR‘𝑇)))
42	39, 41	syl 17	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑉‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) = (ran (2^nd ‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) ∩ (mVR‘𝑇)))
43		fvex 6708	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑋) ∈ V
44		fvex 6708	. . . . . . . . . . . . 13 ⊢ (𝑓‘(2^nd ‘𝑋)) ∈ V
45	43, 44	op2nd 7748	. . . . . . . . . . . 12 ⊢ (2^nd ‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) = (𝑓‘(2^nd ‘𝑋))
46	45	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (2^nd ‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) = (𝑓‘(2^nd ‘𝑋)))
47	46	rneqd 5792	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ran (2^nd ‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) = ran (𝑓‘(2^nd ‘𝑋)))
48	47	ineq1d 4112	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (ran (2^nd ‘⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘(2^nd ‘𝑋)) ∩ (mVR‘𝑇)))
49	29, 42, 48	3eqtrd 2775	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋)) = (ran (𝑓‘(2^nd ‘𝑋)) ∩ (mVR‘𝑇)))
50	19, 1, 40	mvrsval 33134	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐸 → (𝑉‘𝑋) = (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)))
51	12, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑉‘𝑋) = (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)))
52	51	iuneq1d 4917	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))) = ∪ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))(𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))))
53		msubvrs.h	. . . . . . . . . . . . . . . . 17 ⊢ 𝐻 = (mVH‘𝑇)
54	19, 1, 53	mvhf 33187	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
55	54	3ad2ant1 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝐻:(mVR‘𝑇)⟶𝐸)
56		inss2 4130	. . . . . . . . . . . . . . . 16 ⊢ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
57	56	sseli 3883	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58		ffvelrn 6880	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(mVR‘𝑇)⟶𝐸 ∧ 𝑥 ∈ (mVR‘𝑇)) → (𝐻‘𝑥) ∈ 𝐸)
59	55, 57, 58	syl2an 599	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝐻‘𝑥) ∈ 𝐸)
60		fveq2 6695	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (1^st ‘𝑒) = (1^st ‘(𝐻‘𝑥)))
61		2fveq3 6700	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘(𝐻‘𝑥))))
62	60, 61	opeq12d 4778	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = (𝐻‘𝑥) → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘(𝐻‘𝑥)), (𝑓‘(2^nd ‘(𝐻‘𝑥)))⟩)
63	62, 25, 26	fvmpt3i 6801	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑥) ∈ 𝐸 → ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥)) = ⟨(1^st ‘(𝐻‘𝑥)), (𝑓‘(2^nd ‘(𝐻‘𝑥)))⟩)
64	59, 63	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥)) = ⟨(1^st ‘(𝐻‘𝑥)), (𝑓‘(2^nd ‘(𝐻‘𝑥)))⟩)
65	57	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
66		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (mType‘𝑇) = (mType‘𝑇)
67	19, 66, 53	mvhval 33163	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (mVR‘𝑇) → (𝐻‘𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩)
68	65, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝐻‘𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩)
69		fvex 6708	. . . . . . . . . . . . . . . 16 ⊢ ((mType‘𝑇)‘𝑥) ∈ V
70		s1cli 14127	. . . . . . . . . . . . . . . . 17 ⊢ ⟨“𝑥”⟩ ∈ Word V
71	70	elexi 3417	. . . . . . . . . . . . . . . 16 ⊢ ⟨“𝑥”⟩ ∈ V
72	69, 71	op1std 7749	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩ → (1^st ‘(𝐻‘𝑥)) = ((mType‘𝑇)‘𝑥))
73	68, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (1^st ‘(𝐻‘𝑥)) = ((mType‘𝑇)‘𝑥))
74	69, 71	op2ndd 7750	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻‘𝑥) = ⟨((mType‘𝑇)‘𝑥), ⟨“𝑥”⟩⟩ → (2^nd ‘(𝐻‘𝑥)) = ⟨“𝑥”⟩)
75	68, 74	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (2^nd ‘(𝐻‘𝑥)) = ⟨“𝑥”⟩)
76	75	fveq2d 6699	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑓‘(2^nd ‘(𝐻‘𝑥))) = (𝑓‘⟨“𝑥”⟩))
77	73, 76	opeq12d 4778	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨(1^st ‘(𝐻‘𝑥)), (𝑓‘(2^nd ‘(𝐻‘𝑥)))⟩ = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
78	64, 77	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥)) = ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩)
79	78	fveq2d 6699	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))) = (𝑉‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩))
80		simpl1 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
81	19, 13, 66	mtyf2 33180	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇))
82	80, 81	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇))
83	82, 65	ffvelrnd 6883	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
84	33	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
85		elun2 4077	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (mVR‘𝑇) → 𝑥 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
86	65, 85	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
87	86	s1cld 14125	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
88		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
89	88, 19, 14	mrexval 33130	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ mFS → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
90	80, 89	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
91	87, 90	eleqtrrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
92	84, 91	ffvelrnd 6883	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93		opelxpi 5573	. . . . . . . . . . . . . 14 ⊢ ((((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇) ∧ (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇)) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
94	83, 92, 93	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
95	94, 15	eleqtrrdi 2842	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸)
96	19, 1, 40	mvrsval 33134	. . . . . . . . . . . 12 ⊢ (⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ 𝐸 → (𝑉‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (ran (2^nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) ∩ (mVR‘𝑇)))
97	95, 96	syl 17	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑉‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (ran (2^nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) ∩ (mVR‘𝑇)))
98		fvex 6708	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘⟨“𝑥”⟩) ∈ V
99	69, 98	op2nd 7748	. . . . . . . . . . . . . 14 ⊢ (2^nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (𝑓‘⟨“𝑥”⟩)
100	99	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (2^nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = (𝑓‘⟨“𝑥”⟩))
101	100	rneqd 5792	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ran (2^nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) = ran (𝑓‘⟨“𝑥”⟩))
102	101	ineq1d 4112	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (ran (2^nd ‘⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩) ∩ (mVR‘𝑇)) = (ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
103	79, 97, 102	3eqtrd 2775	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))) = (ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
104	103	iuneq2dv 4914	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ∪ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))(𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))) = ∪ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
105	52, 104	eqtrd 2771	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))) = ∪ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))(ran (𝑓‘⟨“𝑥”⟩) ∩ (mVR‘𝑇)))
106	21, 49, 105	3eqtr4d 2781	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))))
107		fveq1 6694	. . . . . . . . 9 ⊢ (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝐹‘𝑋) = ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋))
108	107	fveq2d 6699	. . . . . . . 8 ⊢ (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝑉‘(𝐹‘𝑋)) = (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋)))
109		fveq1 6694	. . . . . . . . . 10 ⊢ (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝐹‘(𝐻‘𝑥)) = ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥)))
110	109	fveq2d 6699	. . . . . . . . 9 ⊢ (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝑉‘(𝐹‘(𝐻‘𝑥))) = (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))))
111	110	iuneq2d 4919	. . . . . . . 8 ⊢ (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥))))
112	108, 111	eqeq12d 2752	. . . . . . 7 ⊢ (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → ((𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))) ↔ (𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)‘(𝐻‘𝑥)))))
113	106, 112	syl5ibrcom 250	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥)))))
114	113	3expia 1123	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇)) → (𝑋 ∈ 𝐸 → (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))))
115	114	com23 86	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇)) → (𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝑋 ∈ 𝐸 → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))))
116	115	rexlimdva 3193	. . 3 ⊢ (𝑇 ∈ mFS → (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) → (𝑋 ∈ 𝐸 → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))))
117	10, 116	syl5bi 245	. 2 ⊢ (𝑇 ∈ mFS → (𝐹 ∈ ran 𝑆 → (𝑋 ∈ 𝐸 → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))))
118	117	3imp 1113	1 ⊢ ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝐸) → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ∪ cun 3851 ∩ cin 3852 ⟨cop 4533 ∪ ciun 4890 ↦ cmpt 5120 × cxp 5534 ran crn 5537 ⟶wf 6354 ‘cfv 6358 1^st c1st 7737 2^nd c2nd 7738 Word cword 14034 ⟨“cs1 14117 mCNcmcn 33089 mVRcmvar 33090 mTypecmty 33091 mTCcmtc 33093 mRExcmrex 33095 mExcmex 33096 mVarscmvrs 33098 mRSubstcmrsub 33099 mSubstcmsub 33100 mVHcmvh 33101 mFScmfs 33105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-word 14035 df-lsw 14083 df-concat 14091 df-s1 14118 df-substr 14171 df-pfx 14201 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-gsum 16901 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-frmd 18230 df-mrex 33115 df-mex 33116 df-mvrs 33118 df-mrsub 33119 df-msub 33120 df-mvh 33121 df-mfs 33125

This theorem is referenced by: mclsppslem 33212

W3C validator