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
Definition	df-mfs 35501*	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 35502*	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 35503*	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 35504	Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
Theorem	mvtval 35505	The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑉 = (mVT‘𝑇)    &   ⊢ 𝑌 = (mType‘𝑇)    ⇒   ⊢ 𝑉 = ran 𝑌
Theorem	mrexval 35506	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 35507	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 35508	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 35509	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 35510	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 35511	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 35512*	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 35513*	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 35514*	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 35515	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 35516	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 35517	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 35518	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 35519	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 35520	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 35521	The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑆 = (mRSubst‘𝑇)    ⇒   ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
Theorem	mrsubf 35522	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 35523	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 35524	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 35525*	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 35554.) (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 35526	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 35527*	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 35528*	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 35529*	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 35530	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 35531	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 35532	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 35533*	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 35534	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 35535	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 35536	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 35537	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 35538*	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 35539	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 35540*	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)
Theorem	elmpst 35541	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 ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))
Theorem	msrfval 35542*	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 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
Theorem	msrval 35543	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⟩)
Theorem	mpstssv 35544	A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    ⇒   ⊢ 𝑃 ⊆ ((V × V) × V)
Theorem	mpst123 35545	Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    ⇒   ⊢ (𝑋 ∈ 𝑃 → 𝑋 = ⟨(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)⟩)
Theorem	mpstrcl 35546	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))
Theorem	msrf 35547	The reduct of a pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝑅 = (mStRed‘𝑇)    ⇒   ⊢ 𝑅:𝑃⟶𝑃
Theorem	msrrcl 35548	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)    ⇒   ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))
Theorem	mstaval 35549	Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ 𝑆 = ran 𝑅
Theorem	msrid 35550	The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ (𝑋 ∈ 𝑆 → (𝑅‘𝑋) = 𝑋)
Theorem	msrfo 35551	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:𝑃–onto→𝑆
Theorem	mstapst 35552	A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ 𝑆 ⊆ 𝑃
Theorem	elmsta 35553	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⟩ ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍)))
Theorem	ismfs 35554*	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)    ⇒   ⊢ (𝑇 ∈ 𝑊 → (𝑇 ∈ mFS ↔ (((𝐶 ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))))
Theorem	mfsdisj 35555	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 → (𝐶 ∩ 𝑉) = ∅)
Theorem	mtyf2 35556	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:𝑉⟶𝐾)
Theorem	mtyf 35557	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:𝑉⟶𝐹)
Theorem	mvtss 35558	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 → 𝐹 ⊆ 𝐾)
Theorem	maxsta 35559	An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)
Theorem	mvtinf 35560	Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐹 = (mVT‘𝑇)    &   ⊢ 𝑌 = (mType‘𝑇)    ⇒   ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin)
Theorem	msubff1 35561	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 = (mSubst‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    ⇒   ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝐸 ↑m 𝐸))
Theorem	msubff1o 35562	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 = (mSubst‘𝑇)    ⇒   ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆)
Theorem	mvhf 35563	The function mapping variables to variable expressions is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    ⇒   ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
Theorem	mvhf1 35564	The function mapping variables to variable expressions is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    ⇒   ⊢ (𝑇 ∈ mFS → 𝐻:𝑉–1-1→𝐸)
Theorem	msubvrs 35565*	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 = (mSubst‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝑉 = (mVars‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    ⇒   ⊢ ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝐸) → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))
Theorem	mclsrcl 35566	Reverse closure for the closure function. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    ⇒   ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))
Theorem	mclsssvlem 35567*	Lemma for mclsssv 35569. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝑆 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVars‘𝑇)    ⇒   ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸)
Theorem	mclsval 35568*	The function mapping variables to variable expressions is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝑆 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVars‘𝑇)    ⇒   ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
Theorem	mclsssv 35569	The closure of a set of expressions is a set of expressions. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    ⇒   ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝐸)
Theorem	ssmclslem 35570	Lemma for ssmcls 35572. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    ⇒   ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵))
Theorem	vhmcls 35571	All variable hypotheses are in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐻‘𝑋) ∈ (𝐾𝐶𝐵))
Theorem	ssmcls 35572	The original expressions are also in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ (𝐾𝐶𝐵))
Theorem	ss2mcls 35573	The closure is monotonic under subsets of the original set of expressions and the set of disjoint variable conditions. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))
Theorem	mclsax 35574*	The closure is closed under axiom application. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)    ⇒   ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))
Theorem	mclsind 35575*	Induction theorem for closure: any other set 𝑄 closed under the axioms and the hypotheses contains all the elements of the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑄)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄)    &   ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑄)    ⇒   ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)
Theorem	mppspstlem 35576*	Lemma for mppspst 35579. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    ⇒   ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃
Theorem	mppsval 35577*	Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    ⇒   ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
Theorem	elmpps 35578	Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    ⇒   ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))
Theorem	mppspst 35579	A provable pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    ⇒   ⊢ 𝐽 ⊆ 𝑃
Theorem	mthmval 35580	A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. Unlike the difference between pre-statement and statement, this application of the reduct is not necessarily trivial: there are theorems that are not themselves provable but are provable once enough "dummy variables" are introduced. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽))
Theorem	elmthm 35581*	A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
Theorem	mthmi 35582	A statement whose reduct is the reduct of a provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)
Theorem	mthmsta 35583	A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑈 = (mThm‘𝑇)    &   ⊢ 𝑆 = (mPreSt‘𝑇)    ⇒   ⊢ 𝑈 ⊆ 𝑆
Theorem	mppsthm 35584	A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ 𝐽 ⊆ 𝑈
Theorem	mthmblem 35585	Lemma for mthmb 35586. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈))
Theorem	mthmb 35586	If two statements have the same reduct then one is a theorem iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))
Theorem	mthmpps 35587	Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many disjoint variable conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    &   ⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝑉 = (mVars‘𝑇)    &   ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))    &   ⊢ 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))    ⇒   ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
Theorem	mclsppslem 35588*	The closure is closed under application of provable pre-statements. (Compare mclsax 35574.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)    &   ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))    &   ⊢ (𝜑 → 𝑠 ∈ ran 𝐿)    &   ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))    &   ⊢ (𝜑 → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀))    ⇒   ⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
Theorem	mclspps 35589*	The closure is closed under application of provable pre-statements. (Compare mclsax 35574.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)    ⇒   ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))
21.6.14  Grammatical formal systems
Syntax	cm0s 35590	Mapping expressions to statements.
class m0St
Syntax	cmsa 35591	The set of syntax axioms.
class mSA
Syntax	cmwgfs 35592	The set of weakly grammatical formal systems.
class mWGFS
Syntax	cmsy 35593	The syntax typecode function.
class mSyn
Syntax	cmesy 35594	The syntax typecode function for expressions.
class mESyn
Syntax	cmgfs 35595	The set of grammatical formal systems.
class mGFS
Syntax	cmtree 35596	The set of proof trees.
class mTree
Syntax	cmst 35597	The set of syntax trees.
class mST
Syntax	cmsax 35598	The indexing set for a syntax axiom.
class mSAX
Syntax	cmufs 35599	The set of unambiguous formal systems.
class mUFS
Definition	df-m0s 35600	Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)