Theorem mrsubrn 31749

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 31748	. . . . . 6 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑𝑚 𝑅))
5		ffn 6186	. . . . . 6 ⊢ (𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑𝑚 𝑅) → 𝑆 Fn (𝑅 ↑pm 𝑉))
6	4, 5	syl 17	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑pm 𝑉))
7		eleq1w 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ dom 𝑓 ↔ 𝑣 ∈ dom 𝑓))
8		fveq2 6333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝑓‘𝑥) = (𝑓‘𝑣))
9		s1eq 13581	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ⟨“𝑥”⟩ = ⟨“𝑣”⟩)
10	7, 8, 9	ifbieq12d 4253	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
11		eqid 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) = (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))
12		fvex 6343	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑣) ∈ V
13		s1cli 13586	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑣”⟩ ∈ Word V
14	13	elexi 3365	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨“𝑣”⟩ ∈ V
15	12, 14	ifex 4296	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) ∈ V
16	10, 11, 15	fvmpt 6425	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))‘𝑣) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
17	16	adantl 467	. . . . . . . . . . . . . . 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”⟩))
18	17	ifeq1da 4256	. . . . . . . . . . . . . 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”⟩))
19		ifan 4274	. . . . . . . . . . . . . 14 ⊢ if((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓), (𝑓‘𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ 𝑉, if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩), ⟨“𝑣”⟩)
20	18, 19	syl6eqr 2823	. . . . . . . . . . . . 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”⟩))
21		elpmi 8029	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
22	21	adantl 467	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
23	22	simprd 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → dom 𝑓 ⊆ 𝑉)
24	23	sseld 3752	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 → 𝑣 ∈ 𝑉))
25	24	pm4.71rd 546	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑣 ∈ dom 𝑓 ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓)))
26	25	bicomd 213	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓) ↔ 𝑣 ∈ dom 𝑓))
27	26	ifbid 4248	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → if((𝑣 ∈ 𝑉 ∧ 𝑣 ∈ dom 𝑓), (𝑓‘𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩))
28	20, 27	eqtr2d 2806	. . . . . . . . . . . 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”⟩))
29	28	mpteq2dv 4880	. . . . . . . . . . 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”⟩)))
30	29	coeq1d 5423	. . . . . . . . . 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))
31	30	oveq2d 6810	. . . . . . . . 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)))
32	31	mpteq2dv 4880	. . . . . . . 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))))
33		eqid 2771	. . . . . . . . . 10 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
34		eqid 2771	. . . . . . . . . 10 ⊢ (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
35	33, 1, 2, 3, 34	mrsubfval 31744	. . . . . . . . 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))))
36	22, 35	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))))
37	22	simpld 478	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑓:dom 𝑓⟶𝑅)
38	37	adantr 466	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → 𝑓:dom 𝑓⟶𝑅)
39	38	ffvelrnda 6503	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ 𝑅)
40		elun2 3933	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
41	40	ad2antlr 700	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑥 ∈ ((mCN‘𝑇) ∪ 𝑉))
42	41	s1cld 13584	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
43	33, 1, 2	mrexval 31737	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
44	43	ad3antrrr 703	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
45	42, 44	eleqtrrd 2853	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 ∈ dom 𝑓) → ⟨“𝑥”⟩ ∈ 𝑅)
46	39, 45	ifclda 4260	. . . . . . . . . 10 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑥 ∈ 𝑉) → if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩) ∈ 𝑅)
47	46, 11	fmptd 6528	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅)
48		ssid 3774	. . . . . . . . 9 ⊢ 𝑉 ⊆ 𝑉
49	33, 1, 2, 3, 34	mrsubfval 31744	. . . . . . . . 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))))
50	47, 48, 49	sylancl 568	. . . . . . . 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))))
51	32, 36, 50	3eqtr4d 2815	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) = (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))))
52	6	adantr 466	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → 𝑆 Fn (𝑅 ↑pm 𝑉))
53		mapsspm 8044	. . . . . . . . 9 ⊢ (𝑅 ↑𝑚 𝑉) ⊆ (𝑅 ↑pm 𝑉)
54	53	a1i 11	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑅 ↑𝑚 𝑉) ⊆ (𝑅 ↑pm 𝑉))
55		fvex 6343	. . . . . . . . . . 11 ⊢ (mREx‘𝑇) ∈ V
56	2, 55	eqeltri 2846	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
57		fvex 6343	. . . . . . . . . . 11 ⊢ (mVR‘𝑇) ∈ V
58	1, 57	eqeltri 2846	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
59	56, 58	elmap 8039	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) ∈ (𝑅 ↑𝑚 𝑉) ↔ (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)):𝑉⟶𝑅)
60	47, 59	sylibr 224	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩)) ∈ (𝑅 ↑𝑚 𝑉))
61		fnfvima 6640	. . . . . . . 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)))
62	52, 54, 60, 61	syl3anc 1476	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘(𝑥 ∈ 𝑉 ↦ if(𝑥 ∈ dom 𝑓, (𝑓‘𝑥), ⟨“𝑥”⟩))) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
63	51, 62	eqeltrd 2850	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
64	63	ralrimiva 3115	. . . . 5 ⊢ (𝑇 ∈ V → ∀𝑓 ∈ (𝑅 ↑pm 𝑉)(𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
65		ffnfv 6531	. . . . 5 ⊢ (𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑𝑚 𝑉)) ↔ (𝑆 Fn (𝑅 ↑pm 𝑉) ∧ ∀𝑓 ∈ (𝑅 ↑pm 𝑉)(𝑆‘𝑓) ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉))))
66	6, 64, 65	sylanbrc 566	. . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑𝑚 𝑉)))
67		frn 6194	. . . 4 ⊢ (𝑆:(𝑅 ↑pm 𝑉)⟶(𝑆 “ (𝑅 ↑𝑚 𝑉)) → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
68	66, 67	syl 17	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
69		fvprc 6327	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
70	3, 69	syl5eq 2817	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
71	70	rneqd 5492	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ran ∅)
72		rn0 5516	. . . . 5 ⊢ ran ∅ = ∅
73	71, 72	syl6eq 2821	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
74		0ss 4117	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉))
75	73, 74	syl6eqss 3805	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉)))
76	68, 75	pm2.61i 176	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑𝑚 𝑉))
77		imassrn 5619	. 2 ⊢ (𝑆 “ (𝑅 ↑𝑚 𝑉)) ⊆ ran 𝑆
78	76, 77	eqssi 3769	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑𝑚 𝑉))

Syntax hints:  ¬ wn 3   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ∪ cun 3722   ⊆ wss 3724  ∅c0 4064  ifcif 4226   ↦ cmpt 4864  dom cdm 5250  ran crn 5251   “ cima 5253   ∘ ccom 5254   Fn wfn 6027  ⟶wf 6028  ‘cfv 6032  (class class class)co 6794   ↑_𝑚 cmap 8010   ↑_pm cpm 8011  Word cword 13488  ⟨“cs1 13491   Σ_g cgsu 16310  freeMndcfrmd 17593  mCNcmcn 31696  mVRcmvar 31697  mRExcmrex 31702  mRSubstcmrsub 31706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-er 7897  df-map 8012  df-pm 8013  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-card 8966  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-nn 11224  df-2 11282  df-n0 11496  df-z 11581  df-uz 11890  df-fz 12535  df-fzo 12675  df-seq 13010  df-hash 13323  df-word 13496  df-concat 13498  df-s1 13499  df-struct 16067  df-ndx 16068  df-slot 16069  df-base 16071  df-sets 16072  df-ress 16073  df-plusg 16163  df-0g 16311  df-gsum 16312  df-mgm 17451  df-sgrp 17493  df-mnd 17504  df-submnd 17545  df-frmd 17595  df-mrex 31722  df-mrsub 31726

This theorem is referenced by:  mrsubff1o  31751  mrsub0  31752  mrsubccat  31754  mrsubcn  31755  msubrn  31765