Theorem mrsubrn 36278

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 36277	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
5	4	ffnd 6710	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑pm 𝑉))
6		eleq1w 2844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓 ↔ 𝑣 ∈ dom 𝑓))
7		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑓‘𝑥) = (𝑓‘𝑣))
8		s1eq 14747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
9	6, 7, 8	ifbieq12d 4511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
10		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) = (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))
11		fvex 6898	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑣) ∈ V
12		s1cli 14752	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑣”⟩ ∈ Word V
13	12	elexi 3473	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨“𝑣”⟩ ∈ V
14	11, 13	ifex 4533	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) ∈ V
15	9, 10, 14	fvmpt 6993	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
16	15	adantl 487	. . . . . . . . . . . . . . 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 4514	. . . . . . . . . . . . . 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 4536	. . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . 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 8866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
21	20	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
22	21	simprd 501	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → dom 𝑓 ⊆ 𝑉)
23	22	sseld 3930	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 → 𝑣 ∈ 𝑉))
24	23	pm4.71rd 572	. . . . . . . . . . . . . . 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 4506	. . . . . . . . . . . . 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 2797	. . . . . . . . . . . 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 5199	. . . . . . . . . . 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 5839	. . . . . . . . . 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 7436	. . . . . . . . 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 5199	. . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
33		eqid 2761	. . . . . . . . . 10 ⊢ (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
34	32, 1, 2, 3, 33	mrsubfval 36273	. . . . . . . . 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 500	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑓:dom 𝑓⟶𝑅)
37	36	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → 𝑓:dom 𝑓⟶𝑅)
38	37	ffvelcdmda 7084	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ 𝑅)
39		elun2 4129	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
40	39	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
41	40	s1cld 14750	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
42	32, 1, 2	mrexval 36266	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
43	42	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
44	41, 43	eleqtrrd 2864	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
45	38, 44	ifclda 4518	. . . . . . . . . 10 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
46	45	fmpttd 7115	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅)
47		ssid 3953	. . . . . . . . 9 ⊢ 𝑉 ⊆ 𝑉
48	32, 1, 2, 3, 33	mrsubfval 36273	. . . . . . . . 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 598	. . . . . . . 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 2806	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) = (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))))
51	5	adantr 486	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑆 Fn (𝑅 ↑pm 𝑉))
52		mapsspm 8904	. . . . . . . . 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 6899	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
55	1	fvexi 6899	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
56	54, 55	elmap 8899	. . . . . . . . 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 7239	. . . . . . . 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 1398	. . . . . . 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 2861	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
61	60	ralrimiva 3155	. . . . 5 ⊢ (𝑇 ∈ V → ∀𝑓 ∈ (𝑅 ↑pm 𝑉)(𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
62		ffnfv 7119	. . . . 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 595	. . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑m 𝑉)))
64	63	frnd 6718	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
65	3	rnfvprc 6879	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
66		0ss 4350	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
67	65, 66	eqsstrdi 3975	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
68	64, 67	pm2.61i 184	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
69		imassrn 6197	. 2 ⊢ (𝑆 “ (𝑅 ↑m 𝑉)) ⊆ ran 𝑆
70	68, 69	eqssi 3947	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  ifcif 4482   ↦ cmpt 5186  dom cdm 5651  ran crn 5652   “ cima 5654   ∘ ccom 5655   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847   ↑_pm cpm 8848  Word cword 14658  ⟨“cs1 14742   Σ_g cgsu 17611  freeMndcfrmd 19043  mCNcmcn 36225  mVRcmvar 36226  mRExcmrex 36231  mRSubstcmrsub 36235

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-word 14659  df-concat 14716  df-s1 14743  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-gsum 17613  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-frmd 19045  df-mrex 36251  df-mrsub 36255

This theorem is used by:  mrsubff1o  36280  mrsub0  36281  mrsubccat  36283  mrsubcn  36284  msubrn  36294