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Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-gzun 32601 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 32602 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 32603 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 32604* 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 32605 The set of constants.
class mCN
 
Syntaxcmvar 32606 The set of variables.
class mVR
 
Syntaxcmty 32607 The type function.
class mType
 
Syntaxcmvt 32608 The set of variable typecodes.
class mVT
 
Syntaxcmtc 32609 The set of typecodes.
class mTC
 
Syntaxcmax 32610 The set of axioms.
class mAx
 
Syntaxcmrex 32611 The set of raw expressions.
class mREx
 
Syntaxcmex 32612 The set of expressions.
class mEx
 
Syntaxcmdv 32613 The set of distinct variables.
class mDV
 
Syntaxcmvrs 32614 The variables in an expression.
class mVars
 
Syntaxcmrsub 32615 The set of raw substitutions.
class mRSubst
 
Syntaxcmsub 32616 The set of substitutions.
class mSubst
 
Syntaxcmvh 32617 The set of variable hypotheses.
class mVH
 
Syntaxcmpst 32618 The set of pre-statements.
class mPreSt
 
Syntaxcmsr 32619 The reduct of a pre-statement.
class mStRed
 
Syntaxcmsta 32620 The set of statements.
class mStat
 
Syntaxcmfs 32621 The set of formal systems.
class mFS
 
Syntaxcmcls 32622 The closure of a set of statements.
class mCls
 
Syntaxcmpps 32623 The set of provable pre-statements.
class mPPSt
 
Syntaxcmthm 32624 The set of theorems.
class mThm
 
Definitiondf-mcn 32625 Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mCN = Slot 1
 
Definitiondf-mvar 32626 Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVR = Slot 2
 
Definitiondf-mty 32627 Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mType = Slot 3
 
Definitiondf-mtc 32628 Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mTC = Slot 4
 
Definitiondf-mmax 32629 Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mAx = Slot 5
 
Definitiondf-mvt 32630 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 32631 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 32632 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 32633 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 32634* 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 32635* 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 32636* 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 32637* 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 32638* 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 32639* 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 32640 Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
 
Definitiondf-mfs 32641* 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 32642* 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 32643* 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 32644 Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
 
Theoremmvtval 32645 The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVT‘𝑇)    &   𝑌 = (mType‘𝑇)       𝑉 = ran 𝑌
 
Theoremmrexval 32646 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 32647 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 32648 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 32649 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 32650 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 32651 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 32652* 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 32653* 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 32654* 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 32655 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 32656 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 32657 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 32658 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 32659 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 32660 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 32661 The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)       (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
 
Theoremmrsubf 32662 A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑅 = (mREx‘𝑇)       (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
 
Theoremmrsubccat 32663 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 32664 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 32665* 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 32694.) (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 32666 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 32667* 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 32668* 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 32669* 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 32670 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 32671 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 32672 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 32673* 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 32674 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 32675 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 32676 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 32677 A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)       (𝐹 ∈ ran 𝑆𝐹:𝐸𝐸)
 
Theoremmvhfval 32678* 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 32679 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 32680* 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 32681 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 32682* 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 32683 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 32684 A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)       𝑃 ⊆ ((V × V) × V)
 
Theoremmpst123 32685 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)⟩)
 
Theoremmpstrcl 32686 The elements of a pre-statement are sets. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)       (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
 
Theoremmsrf 32687 The reduct of a pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝑅 = (mStRed‘𝑇)       𝑅:𝑃𝑃
 
Theoremmsrrcl 32688 If 𝑋 and 𝑌 have the same reduct, then one is a pre-statement iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝑅 = (mStRed‘𝑇)       ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃𝑌𝑃))
 
Theoremmstaval 32689 Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑆 = (mStat‘𝑇)       𝑆 = ran 𝑅
 
Theoremmsrid 32690 The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑆 = (mStat‘𝑇)       (𝑋𝑆 → (𝑅𝑋) = 𝑋)
 
Theoremmsrfo 32691 The reduct of a pre-statement is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑆 = (mStat‘𝑇)    &   𝑃 = (mPreSt‘𝑇)       𝑅:𝑃onto𝑆
 
Theoremmstapst 32692 A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝑆 = (mStat‘𝑇)       𝑆𝑃
 
Theoremelmsta 32693 Property of being a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝑆 = (mStat‘𝑇)    &   𝑉 = (mVars‘𝑇)    &   𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))       (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑆 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (𝑍 × 𝑍)))
 
Theoremismfs 32694* A formal system is a tuple ⟨mCN, mVR, mType, mVT, mTC, mAx⟩ such that: mCN and mVR are disjoint; mType is a function from mVR to mVT; mVT is a subset of mTC; mAx is a set of statements; and for each variable typecode, there are infinitely many variables of that type. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑌 = (mType‘𝑇)    &   𝐹 = (mVT‘𝑇)    &   𝐾 = (mTC‘𝑇)    &   𝐴 = (mAx‘𝑇)    &   𝑆 = (mStat‘𝑇)       (𝑇𝑊 → (𝑇 ∈ mFS ↔ (((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))))
 
Theoremmfsdisj 32695 The constants and variables of a formal system are disjoint. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)       (𝑇 ∈ mFS → (𝐶𝑉) = ∅)
 
Theoremmtyf2 32696 The type function maps variables to typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐾 = (mTC‘𝑇)    &   𝑌 = (mType‘𝑇)       (𝑇 ∈ mFS → 𝑌:𝑉𝐾)
 
Theoremmtyf 32697 The type function maps variables to variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐹 = (mVT‘𝑇)    &   𝑌 = (mType‘𝑇)       (𝑇 ∈ mFS → 𝑌:𝑉𝐹)
 
Theoremmvtss 32698 The set of variable typecodes is a subset of all typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐹 = (mVT‘𝑇)    &   𝐾 = (mTC‘𝑇)       (𝑇 ∈ mFS → 𝐹𝐾)
 
Theoremmaxsta 32699 An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐴 = (mAx‘𝑇)    &   𝑆 = (mStat‘𝑇)       (𝑇 ∈ mFS → 𝐴𝑆)
 
Theoremmvtinf 32700 Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐹 = (mVT‘𝑇)    &   𝑌 = (mType‘𝑇)       ((𝑇 ∈ mFS ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)
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