P. 356 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cgob 35501	The Godel-set of equivalence.
class ↔𝑔
Syntax	cgoq 35502	The Godel-set of equality.
class =𝑔
Syntax	cgox 35503	The Godel-set of existential quantification. (Note that this is not a wff.)
class ∃𝑔𝑁𝑈
Definition	df-gonot 35504	Define the Godel-set of negation. Here the argument 𝑈 is also a Godel-set corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ¬𝑔𝑈 = (𝑈⊼𝑔𝑈)
Definition	df-goan 35505*	Define the Godel-set of conjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∧𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬𝑔(𝑢⊼𝑔𝑣))
Definition	df-goim 35506*	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 ↦ (𝑢⊼𝑔¬𝑔𝑣))
Definition	df-goor 35507*	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 ↦ (¬𝑔𝑢 →𝑔 𝑣))
Definition	df-gobi 35508*	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 ↦ ((𝑢 →𝑔 𝑣)∧𝑔(𝑣 →𝑔 𝑢)))
Definition	df-goeq 35509*	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 2705 to introduce equality as a defined notion in terms of ∈𝑔. The expression suc (𝑢 ∪ 𝑣) = 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 (𝑢 ∪ 𝑣) / 𝑤⦌∀𝑔𝑤((𝑤∈𝑔𝑢) ↔𝑔 (𝑤∈𝑔𝑣)))
Definition	df-goex 35510	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 = ¬𝑔∀𝑔𝑁¬𝑔𝑈
21.6.12  Models of ZF
Syntax	cgze 35511	The Axiom of Extensionality.
class AxExt
Syntax	cgzr 35512	The Axiom Scheme of Replacement.
class AxRep
Syntax	cgzp 35513	The Axiom of Power Sets.
class AxPow
Syntax	cgzu 35514	The Axiom of Unions.
class AxUn
Syntax	cgzg 35515	The Axiom of Regularity.
class AxReg
Syntax	cgzi 35516	The Axiom of Infinity.
class AxInf
Syntax	cgzf 35517	The set of models of ZF.
class ZF
Definition	df-gzext 35518	The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxExt = (∀𝑔2o((2o∈𝑔∅) ↔𝑔 (2o∈𝑔1o)) →𝑔 (∅=𝑔1o))
Definition	df-gzrep 35519	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) ↦ (∀𝑔3o∃𝑔1o∀𝑔2o(∀𝑔1o𝑢 →𝑔 (2o=𝑔1o)) →𝑔 ∀𝑔1o∀𝑔2o((2o∈𝑔1o) ↔𝑔 ∃𝑔3o((3o∈𝑔∅)∧𝑔∀𝑔1o𝑢))))
Definition	df-gzpow 35520	The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxPow = ∃𝑔1o∀𝑔2o(∀𝑔1o((1o∈𝑔2o) ↔𝑔 (1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
Definition	df-gzun 35521	The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxUn = ∃𝑔1o∀𝑔2o(∃𝑔1o((2o∈𝑔1o)∧𝑔(1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
Definition	df-gzreg 35522	The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxReg = (∃𝑔1o(1o∈𝑔∅) →𝑔 ∃𝑔1o((1o∈𝑔∅)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ¬𝑔(2o∈𝑔∅))))
Definition	df-gzinf 35523	The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ AxInf = ∃𝑔1o((∅∈𝑔1o)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ∃𝑔∅((2o∈𝑔∅)∧𝑔(∅∈𝑔1o))))
Definition	df-gzf 35524*	Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}
21.6.13  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.
Syntax	cmcn 35525	The set of constants.
class mCN
Syntax	cmvar 35526	The set of variables.
class mVR
Syntax	cmty 35527	The type function.
class mType
Syntax	cmvt 35528	The set of variable typecodes.
class mVT
Syntax	cmtc 35529	The set of typecodes.
class mTC
Syntax	cmax 35530	The set of axioms.
class mAx
Syntax	cmrex 35531	The set of raw expressions.
class mREx
Syntax	cmex 35532	The set of expressions.
class mEx
Syntax	cmdv 35533	The set of distinct variables.
class mDV
Syntax	cmvrs 35534	The variables in an expression.
class mVars
Syntax	cmrsub 35535	The set of raw substitutions.
class mRSubst
Syntax	cmsub 35536	The set of substitutions.
class mSubst
Syntax	cmvh 35537	The set of variable hypotheses.
class mVH
Syntax	cmpst 35538	The set of pre-statements.
class mPreSt
Syntax	cmsr 35539	The reduct of a pre-statement.
class mStRed
Syntax	cmsta 35540	The set of statements.
class mStat
Syntax	cmfs 35541	The set of formal systems.
class mFS
Syntax	cmcls 35542	The closure of a set of statements.
class mCls
Syntax	cmpps 35543	The set of provable pre-statements.
class mPPSt
Syntax	cmthm 35544	The set of theorems.
class mThm
Definition	df-mcn 35545	Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mCN = Slot 1
Definition	df-mvar 35546	Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mVR = Slot 2
Definition	df-mty 35547	Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mType = Slot 3
Definition	df-mtc 35548	Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mTC = Slot 4
Definition	df-mmax 35549	Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mAx = Slot 5
Definition	df-mvt 35550	Define the set of variable typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
Definition	df-mrex 35551	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)))
Definition	df-mex 35552	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)))
Definition	df-mdv 35553	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 ))
Definition	df-mvrs 35554*	Define the set of variables in an expression. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))))
Definition	df-mrsub 35555*	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, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
Definition	df-msub 35556*	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) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩)))
Definition	df-mvh 35557*	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”⟩⟩))
Definition	df-mpst 35558*	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)))
Definition	df-msr 35559*	Define the reduct of a pre-statement. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
Definition	df-msta 35560	Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
Definition	df-mfs 35561*	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))}
Definition	df-mcls 35562*	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), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
Definition	df-mpps 35563*	Define the set of provable pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
Definition	df-mthm 35564	Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
Theorem	mvtval 35565	The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑉 = (mVT‘𝑇)    &   ⊢ 𝑌 = (mType‘𝑇)    ⇒   ⊢ 𝑉 = ran 𝑌
Theorem	mrexval 35566	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)    ⇒   ⊢ (𝑇 ∈ 𝑊 → 𝑅 = Word (𝐶 ∪ 𝑉))
Theorem	mexval 35567	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)
Theorem	mexval2 35568	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 (𝐶 ∪ 𝑉))
Theorem	mdvval 35569	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 )
Theorem	mvrsval 35570	The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    ⇒   ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉))
Theorem	mvrsfpw 35571	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)    ⇒   ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin))
Theorem	mrsubffval 35572*	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‘(𝐶 ∪ 𝑉))    ⇒   ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
Theorem	mrsubfval 35573*	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‘(𝐶 ∪ 𝑉))    ⇒   ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
Theorem	mrsubval 35574*	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‘(𝐶 ∪ 𝑉))    ⇒   ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → ((𝑆‘𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
Theorem	mrsubcv 35575	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:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), ⟨“𝑋”⟩))
Theorem	mrsubvr 35576	The value of a substituted variable. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝑅 = (mREx‘𝑇)    &   ⊢ 𝑆 = (mRSubst‘𝑇)    ⇒   ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝑆‘𝐹)‘⟨“𝑋”⟩) = (𝐹‘𝑋))
Theorem	mrsubff 35577	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)    ⇒   ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
Theorem	mrsubrn 35578	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 “ (𝑅 ↑m 𝑉))
Theorem	mrsubff1 35579	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)    ⇒   ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅))
Theorem	mrsubff1o 35580	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)    ⇒   ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆)
Theorem	mrsub0 35581	The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mRSubst‘𝑇)    ⇒   ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
Theorem	mrsubf 35582	A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mRSubst‘𝑇)    &   ⊢ 𝑅 = (mREx‘𝑇)    ⇒   ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝑅⟶𝑅)
Theorem	mrsubccat 35583	Substitution distributes over concatenation. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mRSubst‘𝑇)    &   ⊢ 𝑅 = (mREx‘𝑇)    ⇒   ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))
Theorem	mrsubcn 35584	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 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘⟨“𝑋”⟩) = ⟨“𝑋”⟩)
Theorem	elmrsubrn 35585*	Characterization of the substitutions as functions from expressions to expressions that distribute under concatenation and map constants to themselves. (The constant part uses (𝐶 ∖ 𝑉) because we don't know that 𝐶 and 𝑉 are disjoint until we get to ismfs 35614.) (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mRSubst‘𝑇)    &   ⊢ 𝑅 = (mREx‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐶 = (mCN‘𝑇)    ⇒   ⊢ (𝑇 ∈ 𝑊 → (𝐹 ∈ ran 𝑆 ↔ (𝐹:𝑅⟶𝑅 ∧ ∀𝑐 ∈ (𝐶 ∖ 𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹‘𝑥) ++ (𝐹‘𝑦)))))
Theorem	mrsubco 35586	The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mRSubst‘𝑇)    ⇒   ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)
Theorem	mrsubvrs 35587*	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 𝑆 ∧ 𝑋 ∈ 𝑅) → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
Theorem	msubffval 35588*	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)    ⇒   ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
Theorem	msubfval 35589*	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:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
Theorem	msubval 35590	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:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)
Theorem	msubrsub 35591	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:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘𝑋)))
Theorem	msubty 35592	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:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (1st ‘((𝑆‘𝐹)‘𝑋)) = (1st ‘𝑋))
Theorem	elmsubrn 35593*	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 ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))⟩))
Theorem	msubrn 35594	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 “ (𝑅 ↑m 𝑉))
Theorem	msubff 35595	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)    ⇒   ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
Theorem	msubco 35596	The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mSubst‘𝑇)    ⇒   ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)
Theorem	msubf 35597	A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mSubst‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    ⇒   ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝐸⟶𝐸)
Theorem	mvhfval 35598*	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 = (𝑣 ∈ 𝑉 ↦ ⟨(𝑌‘𝑣), ⟨“𝑣”⟩⟩)
Theorem	mvhval 35599	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)    ⇒   ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = ⟨(𝑌‘𝑋), ⟨“𝑋”⟩⟩)
Theorem	mpstval 35600*	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 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)