Theorem mrsubcv 32365

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 1131	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉))
2	1	s1cld 13801	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ Word (𝐶 ∪ 𝑉))
3		elun 4046	. . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) ↔ (𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉))
4		elfvex 6571	. . . . . . . . 9 ⊢ (𝑋 ∈ (mCN‘𝑇) → 𝑇 ∈ V)
5		mrsubffval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCN‘𝑇)
6	4, 5	eleq2s 2901	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑇 ∈ V)
7		elfvex 6571	. . . . . . . . 9 ⊢ (𝑋 ∈ (mVR‘𝑇) → 𝑇 ∈ V)
8		mrsubffval.v	. . . . . . . . 9 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eleq2s 2901	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑇 ∈ V)
10	6, 9	jaoi 852	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉) → 𝑇 ∈ V)
11	3, 10	sylbi 218	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → 𝑇 ∈ V)
12	11	3ad2ant3 1128	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑇 ∈ V)
13		mrsubffval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
14	5, 8, 13	mrexval 32356	. . . . 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 2886	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ 𝑅)
17		mrsubffval.s	. . . 4 ⊢ 𝑆 = (mRSubst‘𝑇)
18		eqid 2795	. . . 4 ⊢ (freeMnd‘(𝐶 ∪ 𝑉)) = (freeMnd‘(𝐶 ∪ 𝑉))
19	5, 8, 13, 17, 18	mrsubval 32364	. . 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 1402	. 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 1184	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → 𝐹:𝐴⟶𝑅)
22	21	ffvelrnda 6716	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑅)
23	15	ad2antrr 722	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → 𝑅 = Word (𝐶 ∪ 𝑉))
24	22, 23	eleqtrd 2885	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ Word (𝐶 ∪ 𝑉))
25		simplr 765	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → 𝑣 ∈ (𝐶 ∪ 𝑉))
26	25	s1cld 13801	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → ⟨“𝑣”⟩ ∈ Word (𝐶 ∪ 𝑉))
27	24, 26	ifclda 4415	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) ∈ Word (𝐶 ∪ 𝑉))
28	27	fmpttd 6742	. . . . 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 14031	. . . . 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 584	. . . 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 2870	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝑣 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
32		fveq2 6538	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝐹‘𝑣) = (𝐹‘𝑋))
33		s1eq 13798	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
34	31, 32, 33	ifbieq12d 4408	. . . . . . 7 ⊢ (𝑣 = 𝑋 → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
35		eqid 2795	. . . . . . 7 ⊢ (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))
36		fvex 6551	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
37		s1cli 13803	. . . . . . . . 9 ⊢ ⟨“𝑋”⟩ ∈ Word V
38	37	elexi 3456	. . . . . . . 8 ⊢ ⟨“𝑋”⟩ ∈ V
39	36, 38	ifex 4429	. . . . . . 7 ⊢ if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ V
40	34, 35, 39	fvmpt 6635	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
41	40	3ad2ant3 1128	. . . . 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 13799	. . . 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 2831	. . 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 7032	. 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	ffvelrnd 6717	. . . 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 2884	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ Word (𝐶 ∪ 𝑉))
47	5	fvexi 6552	. . . . . . 7 ⊢ 𝐶 ∈ V
48	8	fvexi 6552	. . . . . . 7 ⊢ 𝑉 ∈ V
49	47, 48	unex 7326	. . . . . 6 ⊢ (𝐶 ∪ 𝑉) ∈ V
50		eqid 2795	. . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐶 ∪ 𝑉))) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
51	18, 50	frmdbas 17828	. . . . . 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 2804	. . . 4 ⊢ Word (𝐶 ∪ 𝑉) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
54	53	gsumws1 17815	. . 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 2835	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 396   ∨ wo 842   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∪ cun 3857   ⊆ wss 3859  ifcif 4381   ↦ cmpt 5041   ∘ ccom 5447  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016  Word cword 13707  ⟨“cs1 13793  Basecbs 16312   Σ_g cgsu 16543  freeMndcfrmd 17823  mCNcmcn 32315  mVRcmvar 32316  mRExcmrex 32321  mRSubstcmrsub 32325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-n0 11746  df-z 11830  df-uz 12094  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-word 13708  df-s1 13794  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-plusg 16407  df-0g 16544  df-gsum 16545  df-frmd 17825  df-mrex 32341  df-mrsub 32345

This theorem is referenced by:  mrsubvr  32366  mrsubcn  32374