msubvrs - Mathbox for Mario Carneiro

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Theorem msubvrs 35582

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 2735	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubvrs.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 35550	. . . . 5 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
5	4	eleq2i 2826	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
6		eqid 2735	. . . . 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 6890	. . . . . 6 ⊢ 𝐸 ∈ V
8	7	mptex 7215	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) ∈ V
9	6, 8	elrnmpti 5942	. . . 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 275	. . 3 ⊢ (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
11		simp2 1137	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑓 ∈ ran (mRSubst‘𝑇))
12		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
13		eqid 2735	. . . . . . . . . . . 12 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
14		eqid 2735	. . . . . . . . . . . 12 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
15	13, 1, 14	mexval 35524	. . . . . . . . . . 11 ⊢ 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
16	12, 15	eleqtrdi 2844	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17		xp2nd 8021	. . . . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
20	2, 19, 14	mrsubvrs 35544	. . . . . . . . 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 584	. . . . . . . 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 6876	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (1^st ‘𝑒) = (1^st ‘𝑋))
23		2fveq3 6881	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘𝑋)))
24	22, 23	opeq12d 4857	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑋 → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩)
25		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)
26		opex 5439	. . . . . . . . . . . 12 ⊢ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ ∈ V
27	24, 25, 26	fvmpt3i 6991	. . . . . . . . . . 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 6880	. . . . . . . . 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 8020	. . . . . . . . . . . . 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 35539	. . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . 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	ffvelcdmd 7075	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑓‘(2^nd ‘𝑋)) ∈ (mREx‘𝑇))
37		opelxpi 5691	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
39	38, 15	eleqtrrdi 2845	. . . . . . . . . 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 35527	. . . . . . . . . 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 6889	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑋) ∈ V
44		fvex 6889	. . . . . . . . . . . . 13 ⊢ (𝑓‘(2^nd ‘𝑋)) ∈ V
45	43, 44	op2nd 7997	. . . . . . . . . . . 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 5918	. . . . . . . . . 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 4194	. . . . . . . . 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 2774	. . . . . . . 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 35527	. . . . . . . . . . 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 4995	. . . . . . . . 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 35580	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
55	54	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝐻:(mVR‘𝑇)⟶𝐸)
56		inss2 4213	. . . . . . . . . . . . . . . 16 ⊢ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
57	56	sseli 3954	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58		ffvelcdm 7071	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(mVR‘𝑇)⟶𝐸 ∧ 𝑥 ∈ (mVR‘𝑇)) → (𝐻‘𝑥) ∈ 𝐸)
59	55, 57, 58	syl2an 596	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝐻‘𝑥) ∈ 𝐸)
60		fveq2 6876	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (1^st ‘𝑒) = (1^st ‘(𝐻‘𝑥)))
61		2fveq3 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘(𝐻‘𝑥))))
62	60, 61	opeq12d 4857	. . . . . . . . . . . . . . 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 6991	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
66		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (mType‘𝑇) = (mType‘𝑇)
67	19, 66, 53	mvhval 35556	. . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . 16 ⊢ ((mType‘𝑇)‘𝑥) ∈ V
70		s1cli 14623	. . . . . . . . . . . . . . . . 17 ⊢ ⟨“𝑥”⟩ ∈ Word V
71	70	elexi 3482	. . . . . . . . . . . . . . . 16 ⊢ ⟨“𝑥”⟩ ∈ V
72	69, 71	op1std 7998	. . . . . . . . . . . . . . 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 7999	. . . . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . 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 4857	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 6880	. . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
81	19, 13, 66	mtyf2 35573	. . . . . . . . . . . . . . . 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	ffvelcdmd 7075	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
84	33	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
85		elun2 4158	. . . . . . . . . . . . . . . . . 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 14621	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
88		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
89	88, 19, 14	mrexval 35523	. . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
92	84, 91	ffvelcdmd 7075	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93		opelxpi 5691	. . . . . . . . . . . . . 14 ⊢ ((((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇) ∧ (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇)) → ⟨((mType‘𝑇)‘𝑥), (𝑓‘⟨“𝑥”⟩)⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
94	83, 92, 93	syl2anc 584	. . . . . . . . . . . . 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 2845	. . . . . . . . . . . 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 35527	. . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘⟨“𝑥”⟩) ∈ V
99	69, 98	op2nd 7997	. . . . . . . . . . . . . 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 5918	. . . . . . . . . . . 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 4194	. . . . . . . . . . 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 2774	. . . . . . . . . 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 4992	. . . . . . . . 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 2770	. . . . . . . 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 2780	. . . . . . 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 6875	. . . . . . . . 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 6880	. . . . . . . 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 6875	. . . . . . . . . 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 6880	. . . . . . . . 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 4998	. . . . . . . 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 2751	. . . . . . 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 247	. . . . . 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 1121	. . . . 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 3141	. . 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	biimtrid 242	. 2 ⊢ (𝑇 ∈ mFS → (𝐹 ∈ ran 𝑆 → (𝑋 ∈ 𝐸 → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))))
118	117	3imp 1110	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∃wrex 3060 Vcvv 3459 ∪ cun 3924 ∩ cin 3925 ⟨cop 4607 ∪ ciun 4967 ↦ cmpt 5201 × cxp 5652 ran crn 5655 ⟶wf 6527 ‘cfv 6531 1^st c1st 7986 2^nd c2nd 7987 Word cword 14531 ⟨“cs1 14613 mCNcmcn 35482 mVRcmvar 35483 mTypecmty 35484 mTCcmtc 35486 mRExcmrex 35488 mExcmex 35489 mVarscmvrs 35491 mRSubstcmrsub 35492 mSubstcmsub 35493 mVHcmvh 35494 mFScmfs 35498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-seq 14020 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-0g 17455 df-gsum 17456 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-frmd 18827 df-mrex 35508 df-mex 35509 df-mvrs 35511 df-mrsub 35512 df-msub 35513 df-mvh 35514 df-mfs 35518

This theorem is referenced by: mclsppslem 35605

W3C validator