mrsubcv - Mathbox for Mario Carneiro

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Theorem mrsubcv 34490

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 1139	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉))
2	1	s1cld 14550	. . . 4 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ Word (𝐶 ∪ 𝑉))
3		elun 4148	. . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) ↔ (𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉))
4		elfvex 6927	. . . . . . . . 9 ⊢ (𝑋 ∈ (mCN‘𝑇) → 𝑇 ∈ V)
5		mrsubffval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCN‘𝑇)
6	4, 5	eleq2s 2852	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑇 ∈ V)
7		elfvex 6927	. . . . . . . . 9 ⊢ (𝑋 ∈ (mVR‘𝑇) → 𝑇 ∈ V)
8		mrsubffval.v	. . . . . . . . 9 ⊢ 𝑉 = (mVR‘𝑇)
9	7, 8	eleq2s 2852	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑇 ∈ V)
10	6, 9	jaoi 856	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐶 ∨ 𝑋 ∈ 𝑉) → 𝑇 ∈ V)
11	3, 10	sylbi 216	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → 𝑇 ∈ V)
12	11	3ad2ant3 1136	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → 𝑇 ∈ V)
13		mrsubffval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
14	5, 8, 13	mrexval 34481	. . . . 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 2837	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ⟨“𝑋”⟩ ∈ 𝑅)
17		mrsubffval.s	. . . 4 ⊢ 𝑆 = (mRSubst‘𝑇)
18		eqid 2733	. . . 4 ⊢ (freeMnd‘(𝐶 ∪ 𝑉)) = (freeMnd‘(𝐶 ∪ 𝑉))
19	5, 8, 13, 17, 18	mrsubval 34489	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ ⟨“𝑋”⟩ ∈ 𝑅) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = ((freeMnd‘(𝐶 ∪ 𝑉)) Σ_g ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩)))
20	16, 19	syld3an3 1410	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = ((freeMnd‘(𝐶 ∪ 𝑉)) Σ_g ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩)))
21		simpl1 1192	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → 𝐹:𝐴⟶𝑅)
22	21	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑅)
23	15	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → 𝑅 = Word (𝐶 ∪ 𝑉))
24	22, 23	eleqtrd 2836	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ Word (𝐶 ∪ 𝑉))
25		simplr 768	. . . . . . . 8 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → 𝑣 ∈ (𝐶 ∪ 𝑉))
26	25	s1cld 14550	. . . . . . 7 ⊢ ((((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) ∧ ¬ 𝑣 ∈ 𝐴) → ⟨“𝑣”⟩ ∈ Word (𝐶 ∪ 𝑉))
27	24, 26	ifclda 4563	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) ∧ 𝑣 ∈ (𝐶 ∪ 𝑉)) → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) ∈ Word (𝐶 ∪ 𝑉))
28	27	fmpttd 7112	. . . . 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 14781	. . . . 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 585	. . . 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 2822	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝑣 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
32		fveq2 6889	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝐹‘𝑣) = (𝐹‘𝑋))
33		s1eq 14547	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → ⟨“𝑣”⟩ = ⟨“𝑋”⟩)
34	31, 32, 33	ifbieq12d 4556	. . . . . . 7 ⊢ (𝑣 = 𝑋 → if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
35		eqid 2733	. . . . . . 7 ⊢ (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))
36		fvex 6902	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
37		s1cli 14552	. . . . . . . . 9 ⊢ ⟨“𝑋”⟩ ∈ Word V
38	37	elexi 3494	. . . . . . . 8 ⊢ ⟨“𝑋”⟩ ∈ V
39	36, 38	ifex 4578	. . . . . . 7 ⊢ if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ V
40	34, 35, 39	fvmpt 6996	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 ∪ 𝑉) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))‘𝑋) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
41	40	3ad2ant3 1136	. . . . 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 14548	. . . 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 2773	. . 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 7422	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((freeMnd‘(𝐶 ∪ 𝑉)) Σ_g ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ ⟨“𝑋”⟩)) = ((freeMnd‘(𝐶 ∪ 𝑉)) Σ_g ⟨“if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩)”⟩))
45	28, 1	ffvelcdmd 7085	. . . 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 2835	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ Word (𝐶 ∪ 𝑉))
47	5	fvexi 6903	. . . . . . 7 ⊢ 𝐶 ∈ V
48	8	fvexi 6903	. . . . . . 7 ⊢ 𝑉 ∈ V
49	47, 48	unex 7730	. . . . . 6 ⊢ (𝐶 ∪ 𝑉) ∈ V
50		eqid 2733	. . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐶 ∪ 𝑉))) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
51	18, 50	frmdbas 18730	. . . . . 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 2742	. . . 4 ⊢ Word (𝐶 ∪ 𝑉) = (Base‘(freeMnd‘(𝐶 ∪ 𝑉)))
54	53	gsumws1 18716	. . 3 ⊢ (if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩) ∈ Word (𝐶 ∪ 𝑉) → ((freeMnd‘(𝐶 ∪ 𝑉)) Σ_g ⟨“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)) Σ_g ⟨“if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩)”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
56	20, 44, 55	3eqtrd 2777	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3946 ⊆ wss 3948 ifcif 4528 ↦ cmpt 5231 ∘ ccom 5680 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 Word cword 14461 ⟨“cs1 14542 Basecbs 17141 Σ_g cgsu 17383 freeMndcfrmd 18725 mCNcmcn 34440 mVRcmvar 34441 mRExcmrex 34446 mRSubstcmrsub 34450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-word 14462 df-s1 14543 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-0g 17384 df-gsum 17385 df-frmd 18727 df-mrex 34466 df-mrsub 34470

This theorem is referenced by: mrsubvr 34491 mrsubcn 34499

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