Theorem mrsubcv 35364

Description: The value of a substituted singleton. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mrsubffval.c	⊢ 𝐶 = (mCN‘𝑇)
mrsubffval.v	⊢ 𝑉 = (mVR‘𝑇)
mrsubffval.r	⊢ 𝑅 = (mREx‘𝑇)
mrsubffval.s	⊢ 𝑆 = (mRSubst‘𝑇)

Assertion

Ref	Expression
mrsubcv	⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))

Proof of Theorem mrsubcv

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1135	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉))
2	1	s1cld 14619	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ Word (𝐶 ∪ 𝑉))
3		elun 4155	. . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) ↔ (𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉))
4		elfvex 6940	. . . . . . . . 9 ⊢ (𝑋 ∈ (mCN‘𝑇) → 𝑇 ∈ V)
5		mrsubffval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCN‘𝑇)
6	4, 5	eleq2s 2847	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑇 ∈ V)
7		elfvex 6940	. . . . . . . . 9 ⊢ (𝑋 ∈ (mVR‘𝑇) → 𝑇 ∈ V)
8		mrsubffval.v	. . . . . . . . 9 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eleq2s 2847	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑇 ∈ V)
10	6, 9	jaoi 855	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉) → 𝑇 ∈ V)
11	3, 10	sylbi 216	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → 𝑇 ∈ V)
12	11	3ad2ant3 1132	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑇 ∈ V)
13		mrsubffval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
14	5, 8, 13	mrexval 35355	. . . . 5 ⊢ (𝑇 ∈ V → 𝑅 = Word (𝐶 ∪ 𝑉))
15	12, 14	syl 17	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑅 = Word (𝐶 ∪ 𝑉))
16	2, 15	eleqtrrd 2832	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ 𝑅)
17		mrsubffval.s	. . . 4 ⊢ 𝑆 = (mRSubst‘𝑇)
18		eqid 2729	. . . 4 ⊢ (freeMnd‘(𝐶 ∪ 𝑉)) = (freeMnd‘(𝐶 ∪ 𝑉))
19	5, 8, 13, 17, 18	mrsubval 35363	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ ⟨“𝑋”⟩ ∈ 𝑅) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = ((freeMnd‘(𝐶 ∪ 𝑉)) Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩)))
20	16, 19	syld3an3 1406	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = ((freeMnd‘(𝐶 ∪ 𝑉)) Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩)))
21		simpl1 1188	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → 𝐹:𝐴⟶𝑅)
22	21	ffvelcdmda 7099	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑅)
23	15	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → 𝑅 = Word (𝐶 ∪ 𝑉))
24	22, 23	eleqtrd 2831	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ Word (𝐶 ∪ 𝑉))
25		simplr 767	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → 𝑣 ∈ (𝐶 ∪ 𝑉))
26	25	s1cld 14619	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → ⟨“𝑣”⟩ ∈ Word (𝐶 ∪ 𝑉))
27	24, 26	ifclda 4568	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) ∈ Word (𝐶 ∪ 𝑉))
28	27	fmpttd 7130	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)):(𝐶 ∪ 𝑉)⟶Word (𝐶 ∪ 𝑉))
29		s1co 14850	. . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∪ 𝑉) ∧ (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)):(𝐶 ∪ 𝑉)⟶Word (𝐶 ∪ 𝑉)) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩) = ⟨“((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋)”⟩)
30	1, 28, 29	syl2anc 582	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩) = ⟨“((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋)”⟩)
31		eleq1 2817	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝑣 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
32		fveq2 6902	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝐹‘𝑣) = (𝐹‘𝑋))
33		s1eq 14616	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
34	31, 32, 33	ifbieq12d 4561	. . . . . . 7 ⊢ (𝑣 = 𝑋 → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
35		eqid 2729	. . . . . . 7 ⊢ (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))
36		fvex 6915	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
37		s1cli 14621	. . . . . . . . 9 ⊢ ⟨“𝑋”⟩ ∈ Word V
38	37	elexi 3493	. . . . . . . 8 ⊢ ⟨“𝑋”⟩ ∈ V
39	36, 38	ifex 4583	. . . . . . 7 ⊢ if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ V
40	34, 35, 39	fvmpt 7010	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
41	40	3ad2ant3 1132	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
42	41	s1eqd 14617	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋)”⟩ = ⟨“if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩)”⟩)
43	30, 42	eqtrd 2769	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩) = ⟨“if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩)”⟩)
44	43	oveq2d 7441	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((freeMnd‘(𝐶 ∪ 𝑉)) Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩)) = ((freeMnd‘(𝐶 ∪ 𝑉)) Σg ⟨“if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩)”⟩))
45	28, 1	ffvelcdmd 7100	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋) ∈ Word (𝐶 ∪ 𝑉))
46	41, 45	eqeltrrd 2830	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ Word (𝐶 ∪ 𝑉))
47	5	fvexi 6916	. . . . . . 7 ⊢ 𝐶 ∈ V
48	8	fvexi 6916	. . . . . . 7 ⊢ 𝑉 ∈ V
49	47, 48	unex 7755	. . . . . 6 ⊢ (𝐶 ∪ 𝑉) ∈ V
50		eqid 2729	. . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐶 ∪ 𝑉))) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
51	18, 50	frmdbas 18850	. . . . . 6 ⊢ ((𝐶 ∪ 𝑉) ∈ V → (Base‘(freeMnd‘(𝐶 ∪ 𝑉))) = Word (𝐶 ∪ 𝑉))
52	49, 51	ax-mp 5	. . . . 5 ⊢ (Base‘(freeMnd‘(𝐶 ∪ 𝑉))) = Word (𝐶 ∪ 𝑉)
53	52	eqcomi 2738	. . . 4 ⊢ Word (𝐶 ∪ 𝑉) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
54	53	gsumws1 18836	. . 3 ⊢ (if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ Word (𝐶 ∪ 𝑉) → ((freeMnd‘(𝐶 ∪ 𝑉)) Σg ⟨“if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩)”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
55	46, 54	syl 17	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((freeMnd‘(𝐶 ∪ 𝑉)) Σg ⟨“if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩)”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
56	20, 44, 55	3eqtrd 2773	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  Vcvv 3471   ∪ cun 3954   ⊆ wss 3956  ifcif 4533   ↦ cmpt 5236   ∘ ccom 5687  ⟶wf 6551  ‘cfv 6555  (class class class)co 7425  Word cword 14530  ⟨“cs1 14611  Basecbs 17221   Σ_g cgsu 17463  freeMndcfrmd 18845  mCNcmcn 35314  mVRcmvar 35315  mRExcmrex 35320  mRSubstcmrsub 35324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5370  ax-pr 5434  ax-un 7746  ax-cnex 11218  ax-resscn 11219  ax-1cn 11220  ax-icn 11221  ax-addcl 11222  ax-addrcl 11223  ax-mulcl 11224  ax-mulrcl 11225  ax-mulcom 11226  ax-addass 11227  ax-mulass 11228  ax-distr 11229  ax-i2m1 11230  ax-1ne0 11231  ax-1rid 11232  ax-rnegex 11233  ax-rrecex 11234  ax-cnre 11235  ax-pre-lttri 11236  ax-pre-lttrn 11237  ax-pre-ltadd 11238  ax-pre-mulgt0 11239

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-nel 3040  df-ral 3055  df-rex 3064  df-rmo 3372  df-reu 3373  df-rab 3428  df-v 3473  df-sbc 3786  df-csb 3902  df-dif 3959  df-un 3961  df-in 3963  df-ss 3973  df-pss 3976  df-nul 4333  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-int 4957  df-iun 5005  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5581  df-eprel 5587  df-po 5595  df-so 5596  df-fr 5638  df-we 5640  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6314  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6507  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7381  df-ov 7428  df-oprab 7429  df-mpo 7430  df-om 7880  df-1st 8006  df-2nd 8007  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-1o 8499  df-er 8737  df-map 8860  df-pm 8861  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-card 9986  df-pnf 11304  df-mnf 11305  df-xr 11306  df-ltxr 11307  df-le 11308  df-sub 11500  df-neg 11501  df-nn 12272  df-2 12334  df-n0 12532  df-z 12618  df-uz 12882  df-fz 13546  df-fzo 13689  df-seq 14029  df-hash 14356  df-word 14531  df-s1 14612  df-struct 17157  df-slot 17192  df-ndx 17204  df-base 17222  df-plusg 17287  df-0g 17464  df-gsum 17465  df-frmd 18847  df-mrex 35340  df-mrsub 35344

This theorem is referenced by:  mrsubvr  35365  mrsubcn  35373