Theorem mrsubrn 35903

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 35902	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
5	4	ffnd 6707	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑pm 𝑉))
6		eleq1w 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓 ↔ 𝑣 ∈ dom 𝑓))
7		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑓‘𝑥) = (𝑓‘𝑣))
8		s1eq 14637	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
9	6, 7, 8	ifbieq12d 4521	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
10		eqid 2769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) = (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))
11		fvex 6895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑣) ∈ V
12		s1cli 14642	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑣”⟩ ∈ Word V
13	12	elexi 3485	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨“𝑣”⟩ ∈ V
14	11, 13	ifex 4543	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) ∈ V
15	9, 10, 14	fvmpt 6990	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
16	15	adantl 486	. . . . . . . . . . . . . . 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 4524	. . . . . . . . . . . . . 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 4546	. . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . 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 8842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
21	20	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
22	21	simprd 500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → dom 𝑓 ⊆ 𝑉)
23	22	sseld 3944	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 → 𝑣 ∈ 𝑉))
24	23	pm4.71rd 571	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓)))
25	24	bicomd 226	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓) ↔ 𝑣 ∈ dom 𝑓))
26	25	ifbid 4516	. . . . . . . . . . . . 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 2805	. . . . . . . . . . . 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 5209	. . . . . . . . . . 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 5848	. . . . . . . . . 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 7427	. . . . . . . . 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 5209	. . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
33		eqid 2769	. . . . . . . . . 10 ⊢ (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
34	32, 1, 2, 3, 33	mrsubfval 35898	. . . . . . . . 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 18	. . . . . . . 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 499	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑓:dom 𝑓⟶𝑅)
37	36	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → 𝑓:dom 𝑓⟶𝑅)
38	37	ffvelcdmda 7080	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ 𝑅)
39		elun2 4144	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
40	39	ad2antlr 739	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
41	40	s1cld 14640	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
42	32, 1, 2	mrexval 35891	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
43	42	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
44	41, 43	eleqtrrd 2872	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
45	38, 44	ifclda 4528	. . . . . . . . . 10 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
46	45	fmpttd 7111	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅)
47		ssid 3967	. . . . . . . . 9 ⊢ 𝑉 ⊆ 𝑉
48	32, 1, 2, 3, 33	mrsubfval 35898	. . . . . . . . 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 597	. . . . . . . 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 2814	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) = (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))))
51	5	adantr 485	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑆 Fn (𝑅 ↑pm 𝑉))
52		mapsspm 8873	. . . . . . . . 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 6896	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
55	1	fvexi 6896	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
56	54, 55	elmap 8868	. . . . . . . . 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 237	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) ∈ (𝑅 ↑m 𝑉))
58		fnfvima 7232	. . . . . . . 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 1396	. . . . . . 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 2869	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
61	60	ralrimiva 3163	. . . . 5 ⊢ (𝑇 ∈ V → ∀𝑓 ∈ (𝑅 ↑pm 𝑉)(𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
62		ffnfv 7115	. . . . 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 594	. . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑m 𝑉)))
64	63	frnd 6715	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
65	3	rnfvprc 6876	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
66		0ss 4364	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
67	65, 66	eqsstrdi 3989	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
68	64, 67	pm2.61i 184	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
69		imassrn 6074	. 2 ⊢ (𝑆 “ (𝑅 ↑m 𝑉)) ⊆ ran 𝑆
70	68, 69	eqssi 3961	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉))

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  ifcif 4492   ↦ cmpt 5196  dom cdm 5662  ran crn 5663   “ cima 5665   ∘ ccom 5666   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8823   ↑_pm cpm 8824  Word cword 14549  ⟨“cs1 14632   Σ_g cgsu 17492  freeMndcfrmd 18905  mCNcmcn 35850  mVRcmvar 35851  mRExcmrex 35856  mRSubstcmrsub 35860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-n0 12504  df-z 12591  df-uz 12862  df-fz 13535  df-fzo 13682  df-seq 14037  df-hash 14366  df-word 14550  df-concat 14607  df-s1 14633  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-0g 17493  df-gsum 17494  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-submnd 18841  df-frmd 18907  df-mrex 35876  df-mrsub 35880

This theorem is referenced by:  mrsubff1o  35905  mrsub0  35906  mrsubccat  35908  mrsubcn  35909  msubrn  35919