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Theorem List for Metamath Proof Explorer - 31301-31400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-fmla 31301 Define the predicate which defines the set of valid Godel formulas. The parameter 𝑛 defines the maximum height of the formulas: the set (Fmla‘∅) is all formulas of the form 𝑥 = 𝑦 or 𝑥𝑦 (which in our coding scheme is the set ({∅, 1𝑜} × (ω × ω)); see df-sat 31299 for the full coding scheme), and each extra level adds to the complexity of the formulas in (Fmla‘𝑛). (Fmla‘ω) = 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
 
Syntaxcgon 31302 The Godel-set of negation. (Note that this is not a wff.)
class ¬𝑔𝑈
 
Syntaxcgoa 31303 The Godel-set of conjunction.
class 𝑔
 
Syntaxcgoi 31304 The Godel-set of implication.
class 𝑔
 
Syntaxcgoo 31305 The Godel-set of disjunction.
class 𝑔
 
Syntaxcgob 31306 The Godel-set of equivalence.
class 𝑔
 
Syntaxcgoq 31307 The Godel-set of equality.
class =𝑔
 
Syntaxcgox 31308 The Godel-set of existential quantification. (Note that this is not a wff.)
class 𝑔𝑁𝑈
 
Definitiondf-gonot 31309 Define the Godel-set of negation. Here the argument 𝑈 is also a Godel-set corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
¬𝑔𝑈 = (𝑈𝑔𝑈)
 
Definitiondf-goan 31310* Define the Godel-set of conjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬𝑔(𝑢𝑔𝑣))
 
Definitiondf-goim 31311* Define the Godel-set of implication. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. 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 31312* Define the Godel-set of disjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. 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 31313* Define the Godel-set of equivalence. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. 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 31314* 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 2600 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 31315 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
 
Definitiondf-prv 31316* Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀𝑚 ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚𝑚 ω)}
 
20.5.11  Models of ZF
 
Syntaxcgze 31317 The Axiom of Extensionality.
class AxExt
 
Syntaxcgzr 31318 The Axiom Scheme of Replacement.
class AxRep
 
Syntaxcgzp 31319 The Axiom of Power Sets.
class AxPow
 
Syntaxcgzu 31320 The Axiom of Unions.
class AxUn
 
Syntaxcgzg 31321 The Axiom of Regularity.
class AxReg
 
Syntaxcgzi 31322 The Axiom of Infinity.
class AxInf
 
Syntaxcgzf 31323 The set of models of ZF.
class ZF
 
Definitiondf-gzext 31324 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 31325 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 31326 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 31327 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 31328 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 31329 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 31330* 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))}
 
20.5.12  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 31331 The set of constants.
class mCN
 
Syntaxcmvar 31332 The set of variables.
class mVR
 
Syntaxcmty 31333 The type function.
class mType
 
Syntaxcmvt 31334 The set of variable typecodes.
class mVT
 
Syntaxcmtc 31335 The set of typecodes.
class mTC
 
Syntaxcmax 31336 The set of axioms.
class mAx
 
Syntaxcmrex 31337 The set of raw expressions.
class mREx
 
Syntaxcmex 31338 The set of expressions.
class mEx
 
Syntaxcmdv 31339 The set of distinct variables.
class mDV
 
Syntaxcmvrs 31340 The variables in an expression.
class mVars
 
Syntaxcmrsub 31341 The set of raw substitutions.
class mRSubst
 
Syntaxcmsub 31342 The set of substitutions.
class mSubst
 
Syntaxcmvh 31343 The set of variable hypotheses.
class mVH
 
Syntaxcmpst 31344 The set of pre-statements.
class mPreSt
 
Syntaxcmsr 31345 The reduct of a pre-statement.
class mStRed
 
Syntaxcmsta 31346 The set of statements.
class mStat
 
Syntaxcmfs 31347 The set of formal systems.
class mFS
 
Syntaxcmcls 31348 The closure of a set of statements.
class mCls
 
Syntaxcmpps 31349 The set of provable pre-statements.
class mPPSt
 
Syntaxcmthm 31350 The set of theorems.
class mThm
 
Definitiondf-mcn 31351 Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mCN = Slot 1
 
Definitiondf-mvar 31352 Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVR = Slot 2
 
Definitiondf-mty 31353 Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mType = Slot 3
 
Definitiondf-mtc 31354 Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mTC = Slot 4
 
Definitiondf-mmax 31355 Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mAx = Slot 5
 
Definitiondf-mvt 31356 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 31357 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 31358 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 31359 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 31360* 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 31361* 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 31362* 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 31363* 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 31364* 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 31365* 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 31366 Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
 
Definitiondf-mfs 31367* 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 31368* 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 31369* 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 31370 Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
 
Theoremmvtval 31371 The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVT‘𝑇)    &   𝑌 = (mType‘𝑇)       𝑉 = ran 𝑌
 
Theoremmrexval 31372 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 31373 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 31374 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 31375 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 dv 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 31376 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 31377 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 31378* 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 31379* 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 31380* 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 31381 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 31382 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 31383 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 31384 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 31385 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 31386 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 31387 The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)       (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
 
Theoremmrsubf 31388 A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑅 = (mREx‘𝑇)       (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
 
Theoremmrsubccat 31389 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 31390 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 31391* 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 31420.) (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 31392 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 31393* 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 31394* 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 31395* 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 31396 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 31397 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 31398 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 31399* 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 31400 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))
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