msubvrs - Mathbox for Mario Carneiro

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Theorem msubvrs 35532

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 2729	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubvrs.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 35500	. . . . 5 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
5	4	eleq2i 2820	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)))
6		eqid 2729	. . . . 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 6840	. . . . . 6 ⊢ 𝐸 ∈ V
8	7	mptex 7163	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) ∈ V
9	6, 8	elrnmpti 5908	. . . 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 2729	. . . . . . . . . . . 12 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
14		eqid 2729	. . . . . . . . . . . 12 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
15	13, 1, 14	mexval 35474	. . . . . . . . . . 11 ⊢ 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
16	12, 15	eleqtrdi 2838	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17		xp2nd 7964	. . . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
20	2, 19, 14	mrsubvrs 35494	. . . . . . . . 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 6826	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (1^st ‘𝑒) = (1^st ‘𝑋))
23		2fveq3 6831	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘𝑋)))
24	22, 23	opeq12d 4835	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑋 → ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑋), (𝑓‘(2^nd ‘𝑋))⟩)
25		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩)
26		opex 5411	. . . . . . . . . . . 12 ⊢ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ ∈ V
27	24, 25, 26	fvmpt3i 6939	. . . . . . . . . . 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 6830	. . . . . . . . 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 7963	. . . . . . . . . . . . 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 35489	. . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . 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 7023	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑓‘(2^nd ‘𝑋)) ∈ (mREx‘𝑇))
37		opelxpi 5660	. . . . . . . . . . . 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 2839	. . . . . . . . . 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 35477	. . . . . . . . . 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 6839	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑋) ∈ V
44		fvex 6839	. . . . . . . . . . . . 13 ⊢ (𝑓‘(2^nd ‘𝑋)) ∈ V
45	43, 44	op2nd 7940	. . . . . . . . . . . 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 5884	. . . . . . . . . 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 4172	. . . . . . . . 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 2768	. . . . . . . 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 35477	. . . . . . . . . . 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 4972	. . . . . . . . 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 35530	. . . . . . . . . . . . . . . 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 4191	. . . . . . . . . . . . . . . 16 ⊢ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
57	56	sseli 3933	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58		ffvelcdm 7019	. . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (1^st ‘𝑒) = (1^st ‘(𝐻‘𝑥)))
61		2fveq3 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘(𝐻‘𝑥))))
62	60, 61	opeq12d 4835	. . . . . . . . . . . . . . 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 6939	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . 17 ⊢ (mType‘𝑇) = (mType‘𝑇)
67	19, 66, 53	mvhval 35506	. . . . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . . 16 ⊢ ((mType‘𝑇)‘𝑥) ∈ V
70		s1cli 14530	. . . . . . . . . . . . . . . . 17 ⊢ ⟨“𝑥”⟩ ∈ Word V
71	70	elexi 3461	. . . . . . . . . . . . . . . 16 ⊢ ⟨“𝑥”⟩ ∈ V
72	69, 71	op1std 7941	. . . . . . . . . . . . . . 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 7942	. . . . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . 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 4835	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 6830	. . . . . . . . . . 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 35523	. . . . . . . . . . . . . . . 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 7023	. . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . . . . . . 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 14528	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
88		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
89	88, 19, 14	mrexval 35473	. . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
92	84, 91	ffvelcdmd 7023	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93		opelxpi 5660	. . . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 35477	. . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘⟨“𝑥”⟩) ∈ V
99	69, 98	op2nd 7940	. . . . . . . . . . . . . 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 5884	. . . . . . . . . . . 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 4172	. . . . . . . . . . 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 2768	. . . . . . . . . 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 4969	. . . . . . . . 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 2764	. . . . . . . 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 2774	. . . . . . 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 6825	. . . . . . . . 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 6830	. . . . . . . 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 6825	. . . . . . . . . 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 6830	. . . . . . . . 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 4975	. . . . . . . 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 2745	. . . . . . 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 3130	. . 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 2109 ∃wrex 3053 Vcvv 3438 ∪ cun 3903 ∩ cin 3904 ⟨cop 4585 ∪ ciun 4944 ↦ cmpt 5176 × cxp 5621 ran crn 5624 ⟶wf 6482 ‘cfv 6486 1^st c1st 7929 2^nd c2nd 7930 Word cword 14438 ⟨“cs1 14520 mCNcmcn 35432 mVRcmvar 35433 mTypecmty 35434 mTCcmtc 35436 mRExcmrex 35438 mExcmex 35439 mVarscmvrs 35441 mRSubstcmrsub 35442 mSubstcmsub 35443 mVHcmvh 35444 mFScmfs 35448

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-n0 12403 df-xnn0 12476 df-z 12490 df-uz 12754 df-fz 13429 df-fzo 13576 df-seq 13927 df-hash 14256 df-word 14439 df-lsw 14488 df-concat 14496 df-s1 14521 df-substr 14566 df-pfx 14596 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-0g 17363 df-gsum 17364 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-frmd 18741 df-mrex 35458 df-mex 35459 df-mvrs 35461 df-mrsub 35462 df-msub 35463 df-mvh 35464 df-mfs 35468

This theorem is referenced by: mclsppslem 35555

W3C validator