msubvrs - Mathbox for Mario Carneiro

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Theorem msubvrs 34154

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 34122	. . . . 5 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩))
5	4	eleq2i 2829	. . . 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 6856	. . . . . 6 ⊢ 𝐸 ∈ V
8	7	mptex 7173	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩) ∈ V
9	6, 8	elrnmpti 5915	. . . 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 2736	. . . . . . . . . . . 12 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
14		eqid 2736	. . . . . . . . . . . 12 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
15	13, 1, 14	mexval 34096	. . . . . . . . . . 11 ⊢ 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇))
16	12, 15	eleqtrdi 2848	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
17		xp2nd 7954	. . . . . . . . . 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 34116	. . . . . . . . 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 6842	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (1^st ‘𝑒) = (1^st ‘𝑋))
23		2fveq3 6847	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑋 → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘𝑋)))
24	22, 23	opeq12d 4838	. . . . . . . . . . . 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 5421	. . . . . . . . . . . 12 ⊢ ⟨(1^st ‘𝑒), (𝑓‘(2^nd ‘𝑒))⟩ ∈ V
27	24, 25, 26	fvmpt3i 6953	. . . . . . . . . . 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 6846	. . . . . . . . 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 7953	. . . . . . . . . . . . 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 34111	. . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . 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 7036	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) → (𝑓‘(2^nd ‘𝑋)) ∈ (mREx‘𝑇))
37		opelxpi 5670	. . . . . . . . . . . 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 2849	. . . . . . . . . 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 34099	. . . . . . . . . 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 6855	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑋) ∈ V
44		fvex 6855	. . . . . . . . . . . . 13 ⊢ (𝑓‘(2^nd ‘𝑋)) ∈ V
45	43, 44	op2nd 7930	. . . . . . . . . . . 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 5893	. . . . . . . . . 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 4171	. . . . . . . . 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 2780	. . . . . . . 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 34099	. . . . . . . . . . 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 4981	. . . . . . . . 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 34152	. . . . . . . . . . . . . . . 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 4189	. . . . . . . . . . . . . . . 16 ⊢ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) ⊆ (mVR‘𝑇)
57	56	sseli 3940	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇)) → 𝑥 ∈ (mVR‘𝑇))
58		ffvelcdm 7032	. . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (1^st ‘𝑒) = (1^st ‘(𝐻‘𝑥)))
61		2fveq3 6847	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐻‘𝑥) → (𝑓‘(2^nd ‘𝑒)) = (𝑓‘(2^nd ‘(𝐻‘𝑥))))
62	60, 61	opeq12d 4838	. . . . . . . . . . . . . . 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 6953	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (mType‘𝑇) = (mType‘𝑇)
67	19, 66, 53	mvhval 34128	. . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . 16 ⊢ ((mType‘𝑇)‘𝑥) ∈ V
70		s1cli 14493	. . . . . . . . . . . . . . . . 17 ⊢ ⟨“𝑥”⟩ ∈ Word V
71	70	elexi 3464	. . . . . . . . . . . . . . . 16 ⊢ ⟨“𝑥”⟩ ∈ V
72	69, 71	op1std 7931	. . . . . . . . . . . . . . 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 7932	. . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . 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 4838	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 6846	. . . . . . . . . . 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 34145	. . . . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . . . . . 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 14491	. . . . . . . . . . . . . . . 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 34095	. . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → ⟨“𝑥”⟩ ∈ (mREx‘𝑇))
92	84, 91	ffvelcdmd 7036	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ 𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑋 ∈ 𝐸) ∧ 𝑥 ∈ (ran (2^nd ‘𝑋) ∩ (mVR‘𝑇))) → (𝑓‘⟨“𝑥”⟩) ∈ (mREx‘𝑇))
93		opelxpi 5670	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . 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 34099	. . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘⟨“𝑥”⟩) ∈ V
99	69, 98	op2nd 7930	. . . . . . . . . . . . . 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 5893	. . . . . . . . . . . 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 4171	. . . . . . . . . . 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 2780	. . . . . . . . . 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 4978	. . . . . . . . 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 2776	. . . . . . . 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 2786	. . . . . . 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 6841	. . . . . . . . 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 6846	. . . . . . . 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 6841	. . . . . . . . . 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 6846	. . . . . . . . 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 4983	. . . . . . . 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 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 3152	. . 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 3073 Vcvv 3445 ∪ cun 3908 ∩ cin 3909 ⟨cop 4592 ∪ ciun 4954 ↦ cmpt 5188 × cxp 5631 ran crn 5634 ⟶wf 6492 ‘cfv 6496 1^st c1st 7919 2^nd c2nd 7920 Word cword 14402 ⟨“cs1 14483 mCNcmcn 34054 mVRcmvar 34055 mTypecmty 34056 mTCcmtc 34058 mRExcmrex 34060 mExcmex 34061 mVarscmvrs 34063 mRSubstcmrsub 34064 mSubstcmsub 34065 mVHcmvh 34066 mFScmfs 34070

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-word 14403 df-lsw 14451 df-concat 14459 df-s1 14484 df-substr 14529 df-pfx 14559 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-0g 17323 df-gsum 17324 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-frmd 18659 df-mrex 34080 df-mex 34081 df-mvrs 34083 df-mrsub 34084 df-msub 34085 df-mvh 34086 df-mfs 34090

This theorem is referenced by: mclsppslem 34177

W3C validator