Theorem mrsubrn 34493

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 34492	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
5	4	ffnd 6716	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑pm 𝑉))
6		eleq1w 2817	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓 ↔ 𝑣 ∈ dom 𝑓))
7		fveq2 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑓‘𝑥) = (𝑓‘𝑣))
8		s1eq 14547	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
9	6, 7, 8	ifbieq12d 4556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
10		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) = (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))
11		fvex 6902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑣) ∈ V
12		s1cli 14552	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑣”⟩ ∈ Word V
13	12	elexi 3494	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨“𝑣”⟩ ∈ V
14	11, 13	ifex 4578	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) ∈ V
15	9, 10, 14	fvmpt 6996	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
16	15	adantl 483	. . . . . . . . . . . . . . 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 4559	. . . . . . . . . . . . . 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 4581	. . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . 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 8837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
21	20	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
22	21	simprd 497	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → dom 𝑓 ⊆ 𝑉)
23	22	sseld 3981	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 → 𝑣 ∈ 𝑉))
24	23	pm4.71rd 564	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓)))
25	24	bicomd 222	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓) ↔ 𝑣 ∈ dom 𝑓))
26	25	ifbid 4551	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 5250	. . . . . . . . . . 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 5860	. . . . . . . . . 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 7422	. . . . . . . . 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 5250	. . . . . . . 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 2733	. . . . . . . . . 10 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
33		eqid 2733	. . . . . . . . . 10 ⊢ (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
34	32, 1, 2, 3, 33	mrsubfval 34488	. . . . . . . . 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 496	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑓:dom 𝑓⟶𝑅)
37	36	adantr 482	. . . . . . . . . . . 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 4177	. . . . . . . . . . . . . 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 14550	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
42	32, 1, 2	mrexval 34481	. . . . . . . . . . . . 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 2837	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
45	38, 44	ifclda 4563	. . . . . . . . . 10 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
46	45	fmpttd 7112	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅)
47		ssid 4004	. . . . . . . . 9 ⊢ 𝑉 ⊆ 𝑉
48	32, 1, 2, 3, 33	mrsubfval 34488	. . . . . . . . 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 587	. . . . . . . 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 2783	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) = (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))))
51	5	adantr 482	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑆 Fn (𝑅 ↑pm 𝑉))
52		mapsspm 8867	. . . . . . . . 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 6903	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
55	1	fvexi 6903	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
56	54, 55	elmap 8862	. . . . . . . . 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 233	. . . . . . . 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 1372	. . . . . . 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 2834	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑m 𝑉)))
61	60	ralrimiva 3147	. . . . 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 584	. . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑m 𝑉)))
64	63	frnd 6723	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
65	3	rnfvprc 6883	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
66		0ss 4396	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
67	65, 66	eqsstrdi 4036	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉)))
68	64, 67	pm2.61i 182	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑m 𝑉))
69		imassrn 6069	. 2 ⊢ (𝑆 “ (𝑅 ↑m 𝑉)) ⊆ ran 𝑆
70	68, 69	eqssi 3998	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉))

Syntax hints:  ¬ wn 3   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  ifcif 4528   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   “ cima 5679   ∘ ccom 5680   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406   ↑_m cmap 8817   ↑_pm cpm 8818  Word cword 14461  ⟨“cs1 14542   Σ_g cgsu 17383  freeMndcfrmd 18725  mCNcmcn 34440  mVRcmvar 34441  mRExcmrex 34446  mRSubstcmrsub 34450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-word 14462  df-concat 14518  df-s1 14543  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-0g 17384  df-gsum 17385  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-submnd 18669  df-frmd 18727  df-mrex 34466  df-mrsub 34470

This theorem is referenced by:  mrsubff1o  34495  mrsub0  34496  mrsubccat  34498  mrsubcn  34499  msubrn  34509