Theorem mrsubcv 34104

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 1138	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉))
2	1	s1cld 14491	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ Word (𝐶 ∪ 𝑉))
3		elun 4108	. . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) ↔ (𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉))
4		elfvex 6880	. . . . . . . . 9 ⊢ (𝑋 ∈ (mCN‘𝑇) → 𝑇 ∈ V)
5		mrsubffval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCN‘𝑇)
6	4, 5	eleq2s 2856	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑇 ∈ V)
7		elfvex 6880	. . . . . . . . 9 ⊢ (𝑋 ∈ (mVR‘𝑇) → 𝑇 ∈ V)
8		mrsubffval.v	. . . . . . . . 9 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eleq2s 2856	. . . . . . . 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 1135	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑇 ∈ V)
13		mrsubffval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
14	5, 8, 13	mrexval 34095	. . . . 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 2841	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ 𝑅)
17		mrsubffval.s	. . . 4 ⊢ 𝑆 = (mRSubst‘𝑇)
18		eqid 2736	. . . 4 ⊢ (freeMnd‘(𝐶 ∪ 𝑉)) = (freeMnd‘(𝐶 ∪ 𝑉))
19	5, 8, 13, 17, 18	mrsubval 34103	. . 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 1409	. 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 1191	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → 𝐹:𝐴⟶𝑅)
22	21	ffvelcdmda 7035	. . . . . . . 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 2840	. . . . . . 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 14491	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → ⟨“𝑣”⟩ ∈ Word (𝐶 ∪ 𝑉))
27	24, 26	ifclda 4521	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) ∈ Word (𝐶 ∪ 𝑉))
28	27	fmpttd 7063	. . . . 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 14722	. . . . 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 2825	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝑣 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
32		fveq2 6842	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝐹‘𝑣) = (𝐹‘𝑋))
33		s1eq 14488	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
34	31, 32, 33	ifbieq12d 4514	. . . . . . 7 ⊢ (𝑣 = 𝑋 → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
35		eqid 2736	. . . . . . 7 ⊢ (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))
36		fvex 6855	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
37		s1cli 14493	. . . . . . . . 9 ⊢ ⟨“𝑋”⟩ ∈ Word V
38	37	elexi 3464	. . . . . . . 8 ⊢ ⟨“𝑋”⟩ ∈ V
39	36, 38	ifex 4536	. . . . . . 7 ⊢ if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ V
40	34, 35, 39	fvmpt 6948	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
41	40	3ad2ant3 1135	. . . . 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 14489	. . . 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 2776	. . 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 7373	. 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 7036	. . . 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 2839	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ Word (𝐶 ∪ 𝑉))
47	5	fvexi 6856	. . . . . . 7 ⊢ 𝐶 ∈ V
48	8	fvexi 6856	. . . . . . 7 ⊢ 𝑉 ∈ V
49	47, 48	unex 7680	. . . . . 6 ⊢ (𝐶 ∪ 𝑉) ∈ V
50		eqid 2736	. . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐶 ∪ 𝑉))) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
51	18, 50	frmdbas 18662	. . . . . 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 2745	. . . 4 ⊢ Word (𝐶 ∪ 𝑉) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
54	53	gsumws1 18648	. . 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 2780	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 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∪ cun 3908   ⊆ wss 3910  ifcif 4486   ↦ cmpt 5188   ∘ ccom 5637  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Word cword 14402  ⟨“cs1 14483  Basecbs 17083   Σ_g cgsu 17322  freeMndcfrmd 18657  mCNcmcn 34054  mVRcmvar 34055  mRExcmrex 34060  mRSubstcmrsub 34064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-s1 14484  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-gsum 17324  df-frmd 18659  df-mrex 34080  df-mrsub 34084

This theorem is referenced by:  mrsubvr  34105  mrsubcn  34113