msubvrs - Mathbox for Mario Carneiro

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Theorem msubvrs 34241

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 2731	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubvrs.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 34209	. . . . 5 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
5	4	eleq2i 2824	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
6		eqid 2731	. . . . 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 6861	. . . . . 6 ⊢ 𝐸 ∈ V
8	7	mptex 7178	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) ∈ V
9	6, 8	elrnmpti 5920	. . . 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 274	. . 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 2731	. . . . . . . . . . . 12 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
14		eqid 2731	. . . . . . . . . . . 12 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
15	13, 1, 14	mexval 34183	. . . . . . . . . . 11 ⊢ 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
16	12, 15	eleqtrdi 2842	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17		xp2nd 7959	. . . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
20	2, 19, 14	mrsubvrs 34203	. . . . . . . . 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 6847	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (1^st ‘𝑒) = (1^st ‘𝑋))
23		2fveq3 6852	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘𝑋)))
24	22, 23	opeq12d 4843	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑋 → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩)
25		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)
26		opex 5426	. . . . . . . . . . . 12 ⊢ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ ∈ V
27	24, 25, 26	fvmpt3i 6958	. . . . . . . . . . 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 6851	. . . . . . . . 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 7958	. . . . . . . . . . . . 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 34198	. . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . 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 7041	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑓‘(2^nd ‘𝑋)) ∈ (mREx‘𝑇))
37		opelxpi 5675	. . . . . . . . . . . 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 2843	. . . . . . . . . 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 34186	. . . . . . . . . 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 6860	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑋) ∈ V
44		fvex 6860	. . . . . . . . . . . . 13 ⊢ (𝑓‘(2^nd ‘𝑋)) ∈ V
45	43, 44	op2nd 7935	. . . . . . . . . . . 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 5898	. . . . . . . . . 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 4176	. . . . . . . . 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 34186	. . . . . . . . . . 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 4986	. . . . . . . . 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 34239	. . . . . . . . . . . . . . . 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 4194	. . . . . . . . . . . . . . . 16 ⊢ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
57	56	sseli 3943	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58		ffvelcdm 7037	. . . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (1^st ‘𝑒) = (1^st ‘(𝐻‘𝑥)))
61		2fveq3 6852	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘(𝐻‘𝑥))))
62	60, 61	opeq12d 4843	. . . . . . . . . . . . . . 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 6958	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑥 ∈ (mVR‘𝑇))
66		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (mType‘𝑇) = (mType‘𝑇)
67	19, 66, 53	mvhval 34215	. . . . . . . . . . . . . . . 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 6860	. . . . . . . . . . . . . . . 16 ⊢ ((mType‘𝑇)‘𝑥) ∈ V
70		s1cli 14505	. . . . . . . . . . . . . . . . 17 ⊢ ⟨“𝑥”⟩ ∈ Word V
71	70	elexi 3465	. . . . . . . . . . . . . . . 16 ⊢ ⟨“𝑥”⟩ ∈ V
72	69, 71	op1std 7936	. . . . . . . . . . . . . . 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 7937	. . . . . . . . . . . . . . . 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 6851	. . . . . . . . . . . . . 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 4843	. . . . . . . . . . . . 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 6851	. . . . . . . . . . 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 1191	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑇 ∈ mFS)
81	19, 13, 66	mtyf2 34232	. . . . . . . . . . . . . . . 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 7041	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ((mType‘𝑇)‘𝑥) ∈ (mTC‘𝑇))
84	33	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → 𝑓:(mREx‘𝑇)⟶(mREx‘𝑇))
85		elun2 4142	. . . . . . . . . . . . . . . . . 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 14503	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
88		eqid 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
89	88, 19, 14	mrexval 34182	. . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
92	84, 91	ffvelcdmd 7041	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93		opelxpi 5675	. . . . . . . . . . . . . 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 2843	. . . . . . . . . . . 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 34186	. . . . . . . . . . . 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 6860	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘⟨“𝑥”⟩) ∈ V
99	69, 98	op2nd 7935	. . . . . . . . . . . . . 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 5898	. . . . . . . . . . . 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 4176	. . . . . . . . . . 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 4983	. . . . . . . . 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 6846	. . . . . . . . 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 6851	. . . . . . . 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 6846	. . . . . . . . . 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 6851	. . . . . . . . 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 4988	. . . . . . . 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 2747	. . . . . . 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 246	. . . . . 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 3148	. . 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 241	. 2 ⊢ (𝑇 ∈ mFS → (𝐹 ∈ ran 𝑆 → (𝑋 ∈ 𝐸 → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))))
118	117	3imp 1111	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 Vcvv 3446 ∪ cun 3911 ∩ cin 3912 ⟨cop 4597 ∪ ciun 4959 ↦ cmpt 5193 × cxp 5636 ran crn 5639 ⟶wf 6497 ‘cfv 6501 1^st c1st 7924 2^nd c2nd 7925 Word cword 14414 ⟨“cs1 14495 mCNcmcn 34141 mVRcmvar 34142 mTypecmty 34143 mTCcmtc 34145 mRExcmrex 34147 mExcmex 34148 mVarscmvrs 34150 mRSubstcmrsub 34151 mSubstcmsub 34152 mVHcmvh 34153 mFScmfs 34157

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 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9884 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-nn 12163 df-2 12225 df-n0 12423 df-xnn0 12495 df-z 12509 df-uz 12773 df-fz 13435 df-fzo 13578 df-seq 13917 df-hash 14241 df-word 14415 df-lsw 14463 df-concat 14471 df-s1 14496 df-substr 14541 df-pfx 14571 df-struct 17030 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-plusg 17160 df-0g 17337 df-gsum 17338 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-submnd 18616 df-frmd 18673 df-mrex 34167 df-mex 34168 df-mvrs 34170 df-mrsub 34171 df-msub 34172 df-mvh 34173 df-mfs 34177

This theorem is referenced by: mclsppslem 34264

W3C validator