Theorem mrsubrn 35481

Description: Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mrsubvr.v	⊢ 𝑉 = (mVR‘𝑇)
mrsubvr.r	⊢ 𝑅 = (mREx‘𝑇)
mrsubvr.s	⊢ 𝑆 = (mRSubst‘𝑇)

Assertion

Ref	Expression
mrsubrn	⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉))

Proof of Theorem mrsubrn

Dummy variables 𝑒 𝑓 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrsubvr.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
2		mrsubvr.r	. . . . . . 7 ⊢ 𝑅 = (mREx‘𝑇)
3		mrsubvr.s	. . . . . . 7 ⊢ 𝑆 = (mRSubst‘𝑇)
4	1, 2, 3	mrsubff 35480	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
5	4	ffnd 6748	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑pm 𝑉))
6		eleq1w 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓 ↔ 𝑣 ∈ dom 𝑓))
7		fveq2 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑓‘𝑥) = (𝑓‘𝑣))
8		s1eq 14648	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
9	6, 7, 8	ifbieq12d 4576	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
10		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) = (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))
11		fvex 6933	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑣) ∈ V
12		s1cli 14653	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑣”⟩ ∈ Word V
13	12	elexi 3511	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨“𝑣”⟩ ∈ V
14	11, 13	ifex 4598	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) ∈ V
15	9, 10, 14	fvmpt 7029	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
16	15	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑣 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
17	16	ifeq1da 4579	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ 𝑉, if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩), ⟨“𝑣”⟩))
18		ifan 4601	. . . . . . . . . . . . . 14 ⊢ if((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓), (𝑓‘𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ 𝑉, if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩), ⟨“𝑣”⟩)
19	17, 18	eqtr4di 2798	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩) = if((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓), (𝑓‘𝑣), ⟨“𝑣”⟩))
20		elpmi 8904	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
21	20	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
22	21	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → dom 𝑓 ⊆ 𝑉)
23	22	sseld 4007	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 → 𝑣 ∈ 𝑉))
24	23	pm4.71rd 562	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓)))
25	24	bicomd 223	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓) ↔ 𝑣 ∈ dom 𝑓))
26	25	ifbid 4571	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → if((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓), (𝑓‘𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
27	19, 26	eqtr2d 2781	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩))
28	27	mpteq2dv 5268	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)))
29	28	coeq1d 5886	. . . . . . . . . 10 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))
30	29	oveq2d 7464	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))
31	30	mpteq2dv 5268	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
32		eqid 2740	. . . . . . . . . 10 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
33		eqid 2740	. . . . . . . . . 10 ⊢ (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
34	32, 1, 2, 3, 33	mrsubfval 35476	. . . . . . . . 9 ⊢ ((𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉) → (𝑆‘𝑓) = (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
35	21, 34	syl 17	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) = (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
36	21	simpld 494	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑓:dom 𝑓⟶𝑅)
37	36	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → 𝑓:dom 𝑓⟶𝑅)
38	37	ffvelcdmda 7118	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ 𝑅)
39		elun2 4206	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
40	39	ad2antlr 726	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
41	40	s1cld 14651	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
42	32, 1, 2	mrexval 35469	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
43	42	ad3antrrr 729	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
44	41, 43	eleqtrrd 2847	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
45	38, 44	ifclda 4583	. . . . . . . . . 10 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
46	45	fmpttd 7149	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅)
47		ssid 4031	. . . . . . . . 9 ⊢ 𝑉 ⊆ 𝑉
48	32, 1, 2, 3, 33	mrsubfval 35476	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉) → (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))) = (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
49	46, 47, 48	sylancl 585	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))) = (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ 𝑉, ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
50	31, 35, 49	3eqtr4d 2790	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) = (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))))
51	5	adantr 480	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑆 Fn (𝑅 ↑pm 𝑉))
52		mapsspm 8934	. . . . . . . . 9 ⊢ (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉)
53	52	a1i 11	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉))
54	2	fvexi 6934	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
55	1	fvexi 6934	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
56	54, 55	elmap 8929	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) ∈ (𝑅 ↑m 𝑉) ↔ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅)
57	46, 56	sylibr 234	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) ∈ (𝑅 ↑m 𝑉))
58		fnfvima 7270	. . . . . . . 8 ⊢ ((𝑆 Fn (𝑅 ↑pm 𝑉) ∧ (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉) ∧ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) ∈ (𝑅 ↑m 𝑉)) → (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
59	51, 53, 57, 58	syl3anc 1371	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
60	50, 59	eqeltrd 2844	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
61	60	ralrimiva 3152	. . . . 5 ⊢ (𝑇 ∈ V → ∀𝑓 ∈ (𝑅 ↑pm 𝑉)(𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
62		ffnfv 7153	. . . . 5 ⊢ (𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑m 𝑉)) ↔ (𝑆 Fn (𝑅 ↑pm 𝑉) ∧ ∀𝑓 ∈ (𝑅 ↑pm 𝑉)(𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉))))
63	5, 61, 62	sylanbrc 582	. . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑m 𝑉)))
64	63	frnd 6755	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
65	3	rnfvprc 6914	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
66		0ss 4423	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
67	65, 66	eqsstrdi 4063	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
68	64, 67	pm2.61i 182	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
69		imassrn 6100	. 2 ⊢ (𝑆 “ (𝑅 ↑m 𝑉)) ⊆ ran 𝑆
70	68, 69	eqssi 4025	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  ifcif 4548   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   “ cima 5703   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884   ↑_pm cpm 8885  Word cword 14562  ⟨“cs1 14643   Σ_g cgsu 17500  freeMndcfrmd 18882  mCNcmcn 35428  mVRcmvar 35429  mRExcmrex 35434  mRSubstcmrsub 35438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-frmd 18884  df-mrex 35454  df-mrsub 35458

This theorem is referenced by:  mrsubff1o  35483  mrsub0  35484  mrsubccat  35486  mrsubcn  35487  msubrn  35497