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Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmdvval 30501 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 30502 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 30503 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 30504* 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 30505* 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 30506* 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 30507 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 30508 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 30509 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 30510 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 30511 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 30512 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 30513 The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)       (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)

Theoremmrsubf 30514 A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mRSubst‘𝑇)    &   𝑅 = (mREx‘𝑇)       (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)

Theoremmrsubccat 30515 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 30516 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 30517* 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 30546.) (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 30518 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 30519* 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 30520* 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 30521* 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 30522 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 30523 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 30524 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 30525* 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 30526 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 30527 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 30528 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 30529 A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑆 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)       (𝐹 ∈ ran 𝑆𝐹:𝐸𝐸)

Theoremmvhfval 30530* 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 30531 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 30532* A pre-statement is an ordered triple, whose first member is a symmetric set of dv 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 30533 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 30534* 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 30535 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 30536 A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)       𝑃 ⊆ ((V × V) × V)

Theoremmpst123 30537 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 30538 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 30539 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 30540 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 30541 Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑆 = (mStat‘𝑇)       𝑆 = ran 𝑅

Theoremmsrid 30542 The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑆 = (mStat‘𝑇)       (𝑋𝑆 → (𝑅𝑋) = 𝑋)

Theoremmsrfo 30543 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 30544 A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝑆 = (mStat‘𝑇)       𝑆𝑃

Theoremelmsta 30545 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 30546* 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 30547 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 30548 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 30549 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 30550 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 30551 An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐴 = (mAx‘𝑇)    &   𝑆 = (mStat‘𝑇)       (𝑇 ∈ mFS → 𝐴𝑆)

Theoremmvtinf 30552 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)

Theoremmsubff1 30553 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 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝐸𝑚 𝐸))

Theoremmsubff1o 30554 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 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1-onto→ran 𝑆)

Theoremmvhf 30555 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:𝑉𝐸)

Theoremmvhf1 30556 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:𝑉1-1𝐸)

Theoremmsubvrs 30557* 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 𝑆𝑋𝐸) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))

Theoremmclsrcl 30558 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 ∧ 𝐾𝐷𝐵𝐸))

Theoremmclsssvlem 30559* Lemma for mclsssv 30561. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)    &   𝐴 = (mAx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝑉 = (mVars‘𝑇)       (𝜑 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝐸)

Theoremmclsval 30560* 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)    &   𝐴 = (mAx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝑉 = (mVars‘𝑇)       (𝜑 → (𝐾𝐶𝐵) = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})

Theoremmclsssv 30561 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)       (𝜑 → (𝐾𝐶𝐵) ⊆ 𝐸)

Theoremssmclslem 30562 Lemma for ssmcls 30564. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)       (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵))

Theoremvhmcls 30563 All variable hypotheses are in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐻𝑋) ∈ (𝐾𝐶𝐵))

Theoremssmcls 30564 The original expressions are also in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)       (𝜑𝐵 ⊆ (𝐾𝐶𝐵))

Theoremss2mcls 30565 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))

Theoremmclsax 30566* The closure is closed under axiom application. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐴 = (mAx‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)       (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))

Theoremmclsind 30567* 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐴 = (mAx‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑𝐵𝑄)    &   ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝑄)    &   ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑄)       (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)

Theoremmppspstlem 30568* Lemma for mppspst 30571. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝐽 = (mPPSt‘𝑇)    &   𝐶 = (mCls‘𝑇)       {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃

Theoremmppsval 30569* 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, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}

Theoremelmpps 30570 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⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))

Theoremmppspst 30571 A provable pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝐽 = (mPPSt‘𝑇)       𝐽𝑃

Theoremmthmval 30572 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 “ (𝑅𝐽))

Theoremelmthm 30573* 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)       (𝑋𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))

Theoremmthmi 30574 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)       ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)

Theoremmthmsta 30575 A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑈 = (mThm‘𝑇)    &   𝑆 = (mPreSt‘𝑇)       𝑈𝑆

Theoremmppsthm 30576 A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐽 = (mPPSt‘𝑇)    &   𝑈 = (mThm‘𝑇)       𝐽𝑈

Theoremmthmblem 30577 Lemma for mthmb 30578. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑈 = (mThm‘𝑇)       ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))

Theoremmthmb 30578 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)       ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))

Theoremmthmpps 30579 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 dv 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⟩))))

Theoremmclsppslem 30580* The closure is closed under application of provable pre-statements. (Compare mclsax 30566.) 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐽 = (mPPSt‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)    &   (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))    &   (𝜑𝑠 ∈ ran 𝐿)    &   (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵)))    &   (𝜑 → ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀))       (𝜑 → (𝑠𝑝) ∈ (𝑆 “ (𝐾𝐶𝐵)))

Theoremmclspps 30581* The closure is closed under application of provable pre-statements. (Compare mclsax 30566.) 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐽 = (mPPSt‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)       (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))

20.5.13  Grammatical formal systems

Syntaxcm0s 30582 Mapping expressions to statements.
class m0St

Syntaxcmsa 30583 The set of syntax axioms.
class mSA

Syntaxcmwgfs 30584 The set of weakly grammatical formal systems.
class mWGFS

Syntaxcmsy 30585 The syntax typecode function.
class mSyn

Syntaxcmesy 30586 The syntax typecode function for expressions.
class mESyn

Syntaxcmgfs 30587 The set of grammatical formal systems.
class mGFS

Syntaxcmtree 30588 The set of proof trees.
class mTree

Syntaxcmst 30589 The set of syntax trees.
class mST

Syntaxcmsax 30590 The indexing set for a syntax axiom.
class mSAX

Syntaxcmufs 30591 The set of unambiguous formal sytems.
class mUFS

Definitiondf-m0s 30592 Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)

Definitiondf-msa 30593* Define the set of syntax axioms. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st𝑎) ∈ (mVT‘𝑡) ∧ Fun ((2nd𝑎) ↾ (mVR‘𝑡)))})

Definitiondf-mwgfs 30594* Define the set of weakly grammatical formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑𝑎((⟨𝑑, , 𝑎⟩ ∈ (mAx‘𝑡) ∧ (1st𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))}

Definitiondf-msyn 30595 Define the syntax typecode function. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSyn = Slot 6

Definitiondf-mtree 30596* Define the set of proof trees. (Contributed by Mario Carneiro, 14-Jul-2016.)
mTree = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑟 ∣ (∀𝑒 ∈ ran (mVH‘𝑡)𝑒𝑟⟨(m0St‘𝑒), ∅⟩ ∧ ∀𝑒 𝑒𝑟⟨((mStRed‘𝑡)‘⟨𝑑, , 𝑒⟩), ∅⟩ ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) → ({(𝑠𝑝)} × X𝑒 ∈ (𝑜 ∪ ((mVH‘𝑡) “ ((mVars‘𝑡) “ (𝑜 ∪ {𝑝}))))(𝑟 “ {(𝑠𝑒)})) ⊆ 𝑟)))}))

Definitiondf-mst 30597 Define the function mapping syntax expressions to syntax trees. (Contributed by Mario Carneiro, 14-Jul-2016.)
mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡))))

Definitiondf-msax 30598* Define the indexing set for a syntax axiom's representation in a tree. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝))))

Definitiondf-mufs 30599 Define the set of unambiguous formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}

20.5.14  Models of formal systems

Syntaxcmuv 30600 The universe of a model.
class mUV

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