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Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcbvmpox 5801* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5802 allows 𝐵 to be a function of 𝑥. (Contributed by NM, 29-Dec-2014.)
𝑧𝐵    &   𝑥𝐷    &   𝑧𝐶    &   𝑤𝐶    &   𝑥𝐸    &   𝑦𝐸    &   (𝑥 = 𝑧𝐵 = 𝐷)    &   ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
 
Theoremcbvmpo 5802* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
𝑧𝐶    &   𝑤𝐶    &   𝑥𝐷    &   𝑦𝐷    &   ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
 
Theoremcbvmpov 5803* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3981, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
(𝑥 = 𝑧𝐶 = 𝐸)    &   (𝑦 = 𝑤𝐸 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
 
Theoremdmoprab 5804* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝜑}
 
Theoremdmoprabss 5805* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
 
Theoremrnoprab 5806* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦𝜑}
 
Theoremrnoprab2 5807* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑}
 
Theoremreldmoprab 5808* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabss 5809* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)
 
Theoremeloprabga 5810* The law of concretion for operation class abstraction. Compare elopab 4138. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
 
Theoremeloprabg 5811* The law of concretion for operation class abstraction. Compare elopab 4138. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
 
Theoremssoprab2i 5812* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
(𝜑𝜓)       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremmpov 5813* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}
 
Theoremmpomptx 5814* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremmpompt 5815* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremmpodifsnif 5816 A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)
(𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵𝐷)
 
Theoremmposnif 5817 A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.)
(𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)
 
Theoremfconstmpo 5818* Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremresoprab 5819* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
 
Theoremresoprab2 5820* Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐶𝐴𝐷𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
 
Theoremresmpo 5821* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
((𝐶𝐴𝐷𝐵) → ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = (𝑥𝐶, 𝑦𝐷𝐸))
 
Theoremfunoprabg 5822* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
(∀𝑥𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
 
Theoremfunoprab 5823* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
∃*𝑧𝜑       Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremfnoprabg 5824* Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
(∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
 
Theoremmpofun 5825* The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       Fun 𝐹
 
Theoremfnoprab 5826* Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
(𝜑 → ∃!𝑧𝜓)       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremffnov 5827* An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
 
Theoremfovcl 5828 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
 
Theoremeqfnov 5829* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))
 
Theoremeqfnov2 5830* Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
 
Theoremfnovim 5831* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.)
(𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
 
Theoremmpo2eqb 5832* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5830. (Contributed by Mario Carneiro, 4-Jan-2017.)
(∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))
 
Theoremrnmpo 5833* The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
 
Theoremreldmmpo 5834* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       Rel dom 𝐹
 
Theoremelrnmpog 5835* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
 
Theoremelrnmpo 5836* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
 
Theoremralrnmpo 5837* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   (𝑧 = 𝐶 → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
 
Theoremrexrnmpo 5838* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   (𝑧 = 𝐶 → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
 
Theoremovid 5839* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝑥𝑅𝑦𝑆) → ((𝑥𝐹𝑦) = 𝑧𝜑))
 
Theoremovidig 5840* The value of an operation class abstraction. Compare ovidi 5841. The condition (𝑥𝑅𝑦𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
∃*𝑧𝜑    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       (𝜑 → (𝑥𝐹𝑦) = 𝑧)
 
Theoremovidi 5841* The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)
((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝑥𝑅𝑦𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))
 
Theoremov 5842* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵) = 𝐶𝜃))
 
Theoremovigg 5843* The value of an operation class abstraction. Compare ovig 5844. The condition (𝑥𝑅𝑦𝑆) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   ∃*𝑧𝜑    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
 
Theoremovig 5844* The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
 
Theoremovmpt4g 5845* Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5456.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
 
Theoremovmpos 5846* Value of a function given by the maps-to notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉) → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅)
 
Theoremov2gf 5847* The value of an operation class abstraction. A version of ovmpog 5857 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐺    &   𝑦𝑆    &   (𝑥 = 𝐴𝑅 = 𝐺)    &   (𝑦 = 𝐵𝐺 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpodxf 5848* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐿)    &   (𝜑𝑆𝑋)    &   𝑥𝜑    &   𝑦𝜑    &   𝑦𝐴    &   𝑥𝐵    &   𝑥𝑆    &   𝑦𝑆       (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpodx 5849* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐿)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐿)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpod 5850* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpox 5851* The value of an operation class abstraction. Variant of ovmpoga 5852 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝑥 = 𝐴𝐷 = 𝐿)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpoga 5852* Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpoa 5853* Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)    &   𝑆 ∈ V       ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpodf 5854* Alternate deduction version of ovmpo 5858, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝐷)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))    &   𝑥𝐹    &   𝑥𝜓    &   𝑦𝐹    &   𝑦𝜓       (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
 
Theoremovmpodv 5855* Alternate deduction version of ovmpo 5858, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝐷)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))       (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
 
Theoremovmpodv2 5856* Alternate deduction version of ovmpo 5858, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝐷)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)       (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
 
Theoremovmpog 5857* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = 𝐴𝑅 = 𝐺)    &   (𝑦 = 𝐵𝐺 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpo 5858* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = 𝐴𝑅 = 𝐺)    &   (𝑦 = 𝐵𝐺 = 𝑆)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)    &   𝑆 ∈ V       ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovi3 5859* The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
(((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → 𝑆 ∈ (𝐻 × 𝐻))    &   (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑅 = 𝑆)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}       (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆)
 
Theoremov6g 5860* The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)
(⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑅 = 𝑆)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑧 = 𝑅)}       (((𝐴𝐺𝐵𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆𝐽) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovg 5861* The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝜏 ∧ (𝑥𝑅𝑦𝑆)) → ∃!𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → ((𝐴𝐹𝐵) = 𝐶𝜃))
 
Theoremovres 5862 The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
((𝐴𝐶𝐵𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
 
Theoremovresd 5863 Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
 
Theoremoprssov 5864 The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)
(((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
 
Theoremfovrn 5865 An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
 
Theoremfovrnda 5866 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)       ((𝜑 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
 
Theoremfovrnd 5867 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑆)       (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)
 
Theoremfnrnov 5868* The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)
(𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)})
 
Theoremfoov 5869* An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
(𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)))
 
Theoremfnovrn 5870 An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)
 
Theoremovelrn 5871* A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
(𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
 
Theoremfunimassov 5872* Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
 
Theoremovelimab 5873* Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥𝐵𝑦𝐶 𝐷 = (𝑥𝐹𝑦)))
 
Theoremovconst2 5874 The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
𝐶 ∈ V       ((𝑅𝐴𝑆𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶)
 
Theoremcaovclg 5875* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)       ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
 
Theoremcaovcld 5876* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
 
Theoremcaovcl 5877* Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)       ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
 
Theoremcaovcomg 5878* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
 
Theoremcaovcomd 5879* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)       (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
 
Theoremcaovcom 5880* Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)       (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
 
Theoremcaovassg 5881* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
 
Theoremcaovassd 5882* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
 
Theoremcaovass 5883* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
 
Theoremcaovcang 5884* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))       ((𝜑 ∧ (𝐴𝑇𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
 
Theoremcaovcand 5885* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
 
Theoremcaovcanrd 5886* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶))
 
Theoremcaovcan 5887* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
𝐶 ∈ V    &   ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))       ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))
 
Theoremcaovordig 5888* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovordid 5889* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovordg 5890* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovordd 5891* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovord2d 5892* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
 
Theoremcaovord3d 5893* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   (𝜑𝐷𝑆)       (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶𝐷𝑅𝐵)))
 
Theoremcaovord 5894* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovord2 5895* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)       (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
 
Theoremcaovord3 5896* Ordering law. (Contributed by NM, 29-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   𝐷 ∈ V       (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶𝐷𝑅𝐵))
 
Theoremcaovdig 5897* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))       ((𝜑 ∧ (𝐴𝐾𝐵𝑆𝐶𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
 
Theoremcaovdid 5898* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
 
Theoremcaovdir2d 5899* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))
 
Theoremcaovdirg 5900* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
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