Theorem mrsubcv 32759

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 1134	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉))
2	1	s1cld 13959	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ Word (𝐶 ∪ 𝑉))
3		elun 4127	. . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) ↔ (𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉))
4		elfvex 6705	. . . . . . . . 9 ⊢ (𝑋 ∈ (mCN‘𝑇) → 𝑇 ∈ V)
5		mrsubffval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCN‘𝑇)
6	4, 5	eleq2s 2933	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑇 ∈ V)
7		elfvex 6705	. . . . . . . . 9 ⊢ (𝑋 ∈ (mVR‘𝑇) → 𝑇 ∈ V)
8		mrsubffval.v	. . . . . . . . 9 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eleq2s 2933	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑇 ∈ V)
10	6, 9	jaoi 853	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉) → 𝑇 ∈ V)
11	3, 10	sylbi 219	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → 𝑇 ∈ V)
12	11	3ad2ant3 1131	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑇 ∈ V)
13		mrsubffval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
14	5, 8, 13	mrexval 32750	. . . . 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 2918	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ 𝑅)
17		mrsubffval.s	. . . 4 ⊢ 𝑆 = (mRSubst‘𝑇)
18		eqid 2823	. . . 4 ⊢ (freeMnd‘(𝐶 ∪ 𝑉)) = (freeMnd‘(𝐶 ∪ 𝑉))
19	5, 8, 13, 17, 18	mrsubval 32758	. . 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 1405	. 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 1187	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → 𝐹:𝐴⟶𝑅)
22	21	ffvelrnda 6853	. . . . . . . 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 2917	. . . . . . 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 13959	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → ⟨“𝑣”⟩ ∈ Word (𝐶 ∪ 𝑉))
27	24, 26	ifclda 4503	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) ∈ Word (𝐶 ∪ 𝑉))
28	27	fmpttd 6881	. . . . 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 14197	. . . . 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 586	. . . 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 2902	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝑣 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
32		fveq2 6672	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝐹‘𝑣) = (𝐹‘𝑋))
33		s1eq 13956	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
34	31, 32, 33	ifbieq12d 4496	. . . . . . 7 ⊢ (𝑣 = 𝑋 → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
35		eqid 2823	. . . . . . 7 ⊢ (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))
36		fvex 6685	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
37		s1cli 13961	. . . . . . . . 9 ⊢ ⟨“𝑋”⟩ ∈ Word V
38	37	elexi 3515	. . . . . . . 8 ⊢ ⟨“𝑋”⟩ ∈ V
39	36, 38	ifex 4517	. . . . . . 7 ⊢ if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ V
40	34, 35, 39	fvmpt 6770	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
41	40	3ad2ant3 1131	. . . . 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 13957	. . . 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 2858	. . 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 7174	. 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 6854	. . . 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 2916	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ Word (𝐶 ∪ 𝑉))
47	5	fvexi 6686	. . . . . . 7 ⊢ 𝐶 ∈ V
48	8	fvexi 6686	. . . . . . 7 ⊢ 𝑉 ∈ V
49	47, 48	unex 7471	. . . . . 6 ⊢ (𝐶 ∪ 𝑉) ∈ V
50		eqid 2823	. . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐶 ∪ 𝑉))) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
51	18, 50	frmdbas 18019	. . . . . 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 2832	. . . 4 ⊢ Word (𝐶 ∪ 𝑉) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
54	53	gsumws1 18004	. . 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 2862	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 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∪ cun 3936   ⊆ wss 3938  ifcif 4469   ↦ cmpt 5148   ∘ ccom 5561  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Word cword 13864  ⟨“cs1 13951  Basecbs 16485   Σ_g cgsu 16716  freeMndcfrmd 18014  mCNcmcn 32709  mVRcmvar 32710  mRExcmrex 32715  mRSubstcmrsub 32719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-word 13865  df-s1 13952  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-plusg 16580  df-0g 16717  df-gsum 16718  df-frmd 18016  df-mrex 32735  df-mrsub 32739

This theorem is referenced by:  mrsubvr  32760  mrsubcn  32768