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Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-goim 32801* Define the Godel-set of implication. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢𝑔¬𝑔𝑣))

Definitiondf-goor 32802* Define the Godel-set of disjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬𝑔𝑢𝑔 𝑣))

Definitiondf-gobi 32803* Define the Godel-set of equivalence. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢𝑔 𝑣)∧𝑔(𝑣𝑔 𝑢)))

Definitiondf-goeq 32804* Define the Godel-set of equality. Here the arguments 𝑥 = ⟨𝑁, 𝑃 correspond to vN and vP , so (∅=𝑔1o) actually means v0 = v1 , not 0 = 1. Here we use the trick mentioned in ax-ext 2770 to introduce equality as a defined notion in terms of 𝑔. The expression suc (𝑢𝑣) = max (𝑢, 𝑣) + 1 here is a convenient way of getting a dummy variable distinct from 𝑢 and 𝑣. (Contributed by Mario Carneiro, 14-Jul-2013.)
=𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ suc (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣)))

Definitiondf-goex 32805 Define the Godel-set of existential quantification. Here 𝑁 ∈ ω corresponds to vN , and 𝑈 represents another formula, and this expression is [∃𝑥𝜑] = ∃𝑔𝑁𝑈 where 𝑥 is the 𝑁-th variable, 𝑈 = [𝜑] is the code for 𝜑. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔𝑁𝑈 = ¬𝑔𝑔𝑁¬𝑔𝑈

20.6.12  Models of ZF

Syntaxcgze 32806 The Axiom of Extensionality.
class AxExt

Syntaxcgzr 32807 The Axiom Scheme of Replacement.
class AxRep

Syntaxcgzp 32808 The Axiom of Power Sets.
class AxPow

Syntaxcgzu 32809 The Axiom of Unions.
class AxUn

Syntaxcgzg 32810 The Axiom of Regularity.
class AxReg

Syntaxcgzi 32811 The Axiom of Infinity.
class AxInf

Syntaxcgzf 32812 The set of models of ZF.
class ZF

Definitiondf-gzext 32813 The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxExt = (∀𝑔2o((2o𝑔∅) ↔𝑔 (2o𝑔1o)) →𝑔 (∅=𝑔1o))

Definitiondf-gzrep 32814 The Godel-set version of the Axiom Scheme of Replacement. Since this is a scheme and not a single axiom, it manifests as a function on wffs, each giving rise to a different axiom. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxRep = (𝑢 ∈ (Fmla‘ω) ↦ (∀𝑔3o𝑔1o𝑔2o(∀𝑔1o𝑢𝑔 (2o=𝑔1o)) →𝑔𝑔1o𝑔2o((2o𝑔1o) ↔𝑔𝑔3o((3o𝑔∅)∧𝑔𝑔1o𝑢))))

Definitiondf-gzpow 32815 The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxPow = ∃𝑔1o𝑔2o(∀𝑔1o((1o𝑔2o) ↔𝑔 (1o𝑔∅)) →𝑔 (2o𝑔1o))

Definitiondf-gzun 32816 The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxUn = ∃𝑔1o𝑔2o(∃𝑔1o((2o𝑔1o)∧𝑔(1o𝑔∅)) →𝑔 (2o𝑔1o))

Definitiondf-gzreg 32817 The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxReg = (∃𝑔1o(1o𝑔∅) →𝑔𝑔1o((1o𝑔∅)∧𝑔𝑔2o((2o𝑔1o) →𝑔 ¬𝑔(2o𝑔∅))))

Definitiondf-gzinf 32818 The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxInf = ∃𝑔1o((∅∈𝑔1o)∧𝑔𝑔2o((2o𝑔1o) →𝑔𝑔∅((2o𝑔∅)∧𝑔(∅∈𝑔1o))))

Definitiondf-gzf 32819* Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)
ZF = {𝑚 ∣ ((Tr 𝑚𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}

20.6.13  Metamath formal systems

This is a formalization of Appendix C of the Metamath book, which describes the mathematical representation of a formal system, of which set.mm (this file) is one.

Syntaxcmcn 32820 The set of constants.
class mCN

Syntaxcmvar 32821 The set of variables.
class mVR

Syntaxcmty 32822 The type function.
class mType

Syntaxcmvt 32823 The set of variable typecodes.
class mVT

Syntaxcmtc 32824 The set of typecodes.
class mTC

Syntaxcmax 32825 The set of axioms.
class mAx

Syntaxcmrex 32826 The set of raw expressions.
class mREx

Syntaxcmex 32827 The set of expressions.
class mEx

Syntaxcmdv 32828 The set of distinct variables.
class mDV

Syntaxcmvrs 32829 The variables in an expression.
class mVars

Syntaxcmrsub 32830 The set of raw substitutions.
class mRSubst

Syntaxcmsub 32831 The set of substitutions.
class mSubst

Syntaxcmvh 32832 The set of variable hypotheses.
class mVH

Syntaxcmpst 32833 The set of pre-statements.
class mPreSt

Syntaxcmsr 32834 The reduct of a pre-statement.
class mStRed

Syntaxcmsta 32835 The set of statements.
class mStat

Syntaxcmfs 32836 The set of formal systems.
class mFS

Syntaxcmcls 32837 The closure of a set of statements.
class mCls

Syntaxcmpps 32838 The set of provable pre-statements.
class mPPSt

Syntaxcmthm 32839 The set of theorems.
class mThm

Definitiondf-mcn 32840 Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mCN = Slot 1

Definitiondf-mvar 32841 Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVR = Slot 2

Definitiondf-mty 32842 Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mType = Slot 3

Definitiondf-mtc 32843 Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mTC = Slot 4

Definitiondf-mmax 32844 Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mAx = Slot 5

Definitiondf-mvt 32845 Define the set of variable typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))

Definitiondf-mrex 32846 Define the set of "raw expressions", which are expressions without a typecode attached. (Contributed by Mario Carneiro, 14-Jul-2016.)
mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)))

Definitiondf-mex 32847 Define the set of expressions, which are strings of constants and variables headed by a typecode constant. (Contributed by Mario Carneiro, 14-Jul-2016.)
mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))

Definitiondf-mdv 32848 Define the set of distinct variable conditions, which are pairs of distinct variables. (Contributed by Mario Carneiro, 14-Jul-2016.)
mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))

Definitiondf-mvrs 32849* Define the set of variables in an expression. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))))

Definitiondf-mrsub 32850* Define a substitution of raw expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.)
mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))

Definitiondf-msub 32851* Define a substitution of expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩)))

Definitiondf-mvh 32852* Define the mapping from variables to their variable hypothesis. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))

Definitiondf-mpst 32853* Define the set of all pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))

Definitiondf-msr 32854* Define the reduct of a pre-statement. (Contributed by Mario Carneiro, 14-Jul-2016.)
mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))

Definitiondf-msta 32855 Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))

Definitiondf-mfs 32856* Define the set of all formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin))}

Definitiondf-mcls 32857* Define the closure of a set of statements relative to a set of disjointness constraints. (Contributed by Mario Carneiro, 14-Jul-2016.)
mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))

Definitiondf-mpps 32858* Define the set of provable pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))})

Definitiondf-mthm 32859 Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))

Theoremmvtval 32860 The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVT‘𝑇)    &   𝑌 = (mType‘𝑇)       𝑉 = ran 𝑌

Theoremmrexval 32861 The set of "raw expressions", which are expressions without a typecode, that is, just sequences of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)       (𝑇𝑊𝑅 = Word (𝐶𝑉))

Theoremmexval 32862 The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐾 = (mTC‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑅 = (mREx‘𝑇)       𝐸 = (𝐾 × 𝑅)

Theoremmexval2 32863 The set of expressions, which are pairs whose first element is a typecode, and whose second element is a list of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐾 = (mTC‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)       𝐸 = (𝐾 × Word (𝐶𝑉))

Theoremmdvval 32864 The set of disjoint variable conditions, which are pairs of distinct variables. (This definition differs from appendix C, which uses unordered pairs instead. We use ordered pairs, but all sets of disjoint variable conditions of interest will be symmetric, so it does not matter.) (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐷 = (mDV‘𝑇)       𝐷 = ((𝑉 × 𝑉) ∖ I )

Theoremmvrsval 32865 The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑊 = (mVars‘𝑇)       (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))

Theoremmvrsfpw 32866 The set of variables in an expression is a finite subset of 𝑉. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑊 = (mVars‘𝑇)       (𝑋𝐸 → (𝑊𝑋) ∈ (𝒫 𝑉 ∩ Fin))

Theoremmrsubffval 32867* The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)    &   𝐺 = (freeMnd‘(𝐶𝑉))       (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))

Theoremmrsubfval 32868* The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)    &   𝐺 = (freeMnd‘(𝐶𝑉))       ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))

Theoremmrsubval 32869* The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)    &   𝐺 = (freeMnd‘(𝐶𝑉))       ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → ((𝑆𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))

Theoremmrsubcv 32870 The value of a substituted singleton. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)       ((𝐹:𝐴𝑅𝐴𝑉𝑋 ∈ (𝐶𝑉)) → ((𝑆𝐹)‘⟨“𝑋”⟩) = if(𝑋𝐴, (𝐹𝑋), ⟨“𝑋”⟩))

Theoremmrsubvr 32871 The value of a substituted variable. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)       ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐴) → ((𝑆𝐹)‘⟨“𝑋”⟩) = (𝐹𝑋))

Theoremmrsubff 32872 A substitution is a function from 𝑅 to 𝑅. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)       (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))

Theoremmrsubrn 32873 Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)       ran 𝑆 = (𝑆 “ (𝑅m 𝑉))

Theoremmrsubff1 32874 When restricted to complete mappings, the substitution-producing function is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)       (𝑇𝑊 → (𝑆 ↾ (𝑅m 𝑉)):(𝑅m 𝑉)–1-1→(𝑅m 𝑅))

Theoremmrsubff1o 32875 When restricted to complete mappings, the substitution-producing function is bijective to the set of all substitutions. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mRSubst‘𝑇)       (𝑇𝑊 → (𝑆 ↾ (𝑅m 𝑉)):(𝑅m 𝑉)–1-1-onto→ran 𝑆)

Theoremmrsub0 32876 The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)       (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)

Theoremmrsubf 32877 A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑅 = (mREx‘𝑇)       (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)

Theoremmrsubccat 32878 Substitution distributes over concatenation. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑅 = (mREx‘𝑇)       ((𝐹 ∈ ran 𝑆𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))

Theoremmrsubcn 32879 A substitution does not change the value of constant substrings. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐶 = (mCN‘𝑇)       ((𝐹 ∈ ran 𝑆𝑋 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑋”⟩) = ⟨“𝑋”⟩)

Theoremelmrsubrn 32880* Characterization of the substitutions as functions from expressions to expressions that distribute under concatenation and map constants to themselves. (The constant part uses (𝐶𝑉) because we don't know that 𝐶 and 𝑉 are disjoint until we get to ismfs 32909.) (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐶 = (mCN‘𝑇)       (𝑇𝑊 → (𝐹 ∈ ran 𝑆 ↔ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))))

Theoremmrsubco 32881 The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)       ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)

Theoremmrsubvrs 32882* The set of variables in a substitution is the union, indexed by the variables in the original expression, of the variables in the substitution to that variable. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)       ((𝐹 ∈ ran 𝑆𝑋𝑅) → (ran (𝐹𝑋) ∩ 𝑉) = 𝑥 ∈ (ran 𝑋𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))

Theoremmsubffval 32883* A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑂 = (mRSubst‘𝑇)       (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝑓)‘(2nd𝑒))⟩)))

Theoremmsubfval 32884* A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑂 = (mRSubst‘𝑇)       ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))

Theoremmsubval 32885 A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑂 = (mRSubst‘𝑇)       ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)

Theoremmsubrsub 32886 A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑂 = (mRSubst‘𝑇)       ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (2nd ‘((𝑆𝐹)‘𝑋)) = ((𝑂𝐹)‘(2nd𝑋)))

Theoremmsubty 32887 The type of a substituted expression is the same as the original type. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)       ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (1st ‘((𝑆𝐹)‘𝑋)) = (1st𝑋))

Theoremelmsubrn 32888* Characterization of substitution in terms of raw substitution, without reference to the generating functions. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐸 = (mEx‘𝑇)    &   𝑂 = (mRSubst‘𝑇)    &   𝑆 = (mSubst‘𝑇)       ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (𝑓‘(2nd𝑒))⟩))

Theoremmsubrn 32889 Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mSubst‘𝑇)       ran 𝑆 = (𝑆 “ (𝑅m 𝑉))

Theoremmsubff 32890 A substitution is a function from 𝐸 to 𝐸. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)       (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝐸m 𝐸))

Theoremmsubco 32891 The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mSubst‘𝑇)       ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)

Theoremmsubf 32892 A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)       (𝐹 ∈ ran 𝑆𝐹:𝐸𝐸)

Theoremmvhfval 32893* Value of the function mapping variables to their corresponding variable expressions. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑌 = (mType‘𝑇)    &   𝐻 = (mVH‘𝑇)       𝐻 = (𝑣𝑉 ↦ ⟨(𝑌𝑣), ⟨“𝑣”⟩⟩)

Theoremmvhval 32894 Value of the function mapping variables to their corresponding variable expressions. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝑌 = (mType‘𝑇)    &   𝐻 = (mVH‘𝑇)       (𝑋𝑉 → (𝐻𝑋) = ⟨(𝑌𝑋), ⟨“𝑋”⟩⟩)

Theoremmpstval 32895* A pre-statement is an ordered triple, whose first member is a symmetric set of disjoint variable conditions, whose second member is a finite set of expressions, and whose third member is an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑃 = (mPreSt‘𝑇)       𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

Theoremelmpst 32896 Property of being a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑃 = (mPreSt‘𝑇)       (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸))

Theoremmsrfval 32897* Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVars‘𝑇)    &   𝑃 = (mPreSt‘𝑇)    &   𝑅 = (mStRed‘𝑇)       𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)

Theoremmsrval 32898 Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVars‘𝑇)    &   𝑃 = (mPreSt‘𝑇)    &   𝑅 = (mStRed‘𝑇)    &   𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))       (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)

Theoremmpstssv 32899 A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)       𝑃 ⊆ ((V × V) × V)

Theoremmpst123 32900 Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)       (𝑋𝑃𝑋 = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩)

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