P. 76 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eloprabga 7501*	The law of concretion for operation class abstraction. Compare elopab 5490. (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) Avoid ax-10 2142, ax-11 2158. (Revised by Wolf Lammen, 15-Oct-2024.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
Theorem	eloprabg 7502*	The law of concretion for operation class abstraction. Compare elopab 5490. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
Theorem	ssoprab2i 7503*	Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Theorem	mpov 7504*	Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}
Theorem	mpomptx 7505*	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 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Theorem	mpompt 7506*	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) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Theorem	mpodifsnif 7507	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}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷)
Theorem	mposnif 7508	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}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶)
Theorem	fconstmpo 7509*	Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Theorem	resoprab 7510*	Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}
Theorem	resoprab2 7511*	Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)})
Theorem	resmpo 7512*	Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸))
Theorem	funoprabg 7513*	"At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
Theorem	funoprab 7514*	"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}
Theorem	fnoprabg 7515*	Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
Theorem	mpofun 7516*	The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof shortened by SN, 23-Jul-2024.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ Fun 𝐹
Theorem	fnoprab 7517*	Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
⊢ (𝜑 → ∃!𝑧𝜓)    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	ffnov 7518*	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) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Theorem	fovcld 7519	Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
Theorem	fovcl 7520	Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.)
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
Theorem	eqfnov 7521*	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) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))
Theorem	eqfnov2 7522*	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 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
Theorem	fnov 7523*	Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
Theorem	mpo2eqb 7524*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 7522. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝐷))
Theorem	rnmpo 7525*	The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
Theorem	reldmmpo 7526*	The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ Rel dom 𝐹
Theorem	elrnmpog 7527*	Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶))
Theorem	elrnmpo 7528*	Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)
Theorem	elimampo 7529*	Membership in the image of an operation. (Contributed by SN, 27-Apr-2025.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))
Theorem	elrnmpores 7530*	Membership in the range of a restricted operation class abstraction. (Contributed by Thierry Arnoux, 25-May-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦)))
Theorem	ralrnmpo 7531*	A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
Theorem	rexrnmpo 7532*	A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))
Theorem	ovid 7533*	The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}    ⇒   ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑))
Theorem	ovidig 7534*	The value of an operation class abstraction. Compare ovidi 7535. The condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ ∃*𝑧𝜑    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}    ⇒   ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧)
Theorem	ovidi 7535*	The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}    ⇒   ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))
Theorem	ov 7536*	The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ 𝐶 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃))
Theorem	ovigg 7537*	The value of an operation class abstraction. Compared with ovig 7538, the condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    &   ⊢ ∃*𝑧𝜑    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Theorem	ovig 7538*	The value of an operation class abstraction (weak version). (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Theorem	ovmpt4g 7539*	Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 6982.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶)
Theorem	ovmpos 7540*	Value of a function given by the maps-to notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉) → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅)
Theorem	ov2gf 7541*	The value of an operation class abstraction. A version of ovmpog 7551 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐺    &   ⊢ Ⅎ𝑦𝑆    &   ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺)    &   ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpodxf 7542*	Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑆    &   ⊢ Ⅎ𝑦𝑆    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpodx 7543*	Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpod 7544*	Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpox 7545*	The value of an operation class abstraction. Variant of ovmpoga 7546 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 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)    &   ⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpoga 7546*	Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpoa 7547*	Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpodf 7548*	Alternate deduction version of ovmpo 7552, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓))    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝐹    &   ⊢ Ⅎ𝑦𝜓    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
Theorem	ovmpodv 7549*	Alternate deduction version of ovmpo 7552, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓))    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
Theorem	ovmpodv2 7550*	Alternate deduction version of ovmpo 7552, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = 𝑆))
Theorem	ovmpog 7551*	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 = 𝐴 → 𝑅 = 𝐺)    &   ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpo 7552*	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 = 𝐴 → 𝑅 = 𝐺)    &   ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpot 7553*	The value of an operation is equal to the value of the same operation expressed in maps-to notation. (Contributed by GG, 16-Mar-2025.) (Revised by GG, 13-Apr-2025.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ (𝑥𝐹𝑦))𝐵) = (𝐴𝐹𝐵))
Theorem	fvmpopr2d 7554*	Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)
⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))    &   ⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)
Theorem	ov3 7555*	The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
⊢ 𝑆 ∈ V    &   ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑅 = 𝑆)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}    ⇒   ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆)
Theorem	ov6g 7556*	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⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)}    ⇒   ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovg 7557*	The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜏 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) → ∃!𝑧𝜑)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}    ⇒   ⊢ ((𝜏 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷)) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃))
Theorem	ovres 7558	The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
Theorem	ovresd 7559	Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
Theorem	oprres 7560*	The restriction of an operation is an operation. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐹:(𝑌 × 𝑌)⟶𝑅)    &   ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆)    ⇒   ⊢ (𝜑 → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Theorem	oprssov 7561	The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)
⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Theorem	fovcdm 7562	An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
Theorem	fovcdmda 7563	An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
Theorem	fovcdmd 7564	An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)
Theorem	fnrnov 7565*	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 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)})
Theorem	foov 7566*	An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)))
Theorem	fnovrn 7567	An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)
Theorem	ovelrn 7568*	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 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)))
Theorem	funimassov 7569*	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 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
Theorem	ovelimab 7570*	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)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦)))
Theorem	ovima0 7571	An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.)
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
Theorem	ovconst2 7572	The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶)
Theorem	oprssdm 7573*	Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)
⊢ ¬ ∅ ∈ 𝑆    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)    ⇒   ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹
Theorem	nssdmovg 7574	The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.)
⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅)
Theorem	ndmovg 7575	The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅)
Theorem	ndmov 7576	The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅)
Theorem	ndmovcl 7577	The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)    &   ⊢ ∅ ∈ 𝑆    ⇒   ⊢ (𝐴𝐹𝐵) ∈ 𝑆
Theorem	ndmovrcl 7578	Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ ¬ ∅ ∈ 𝑆    ⇒   ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
Theorem	ndmovcom 7579	Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Theorem	ndmovass 7580	Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ ¬ ∅ ∈ 𝑆    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Theorem	ndmovdistr 7581	Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ ¬ ∅ ∈ 𝑆    &   ⊢ dom 𝐺 = (𝑆 × 𝑆)    ⇒   ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
Theorem	ndmovord 7582	Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ 𝑅 ⊆ (𝑆 × 𝑆)    &   ⊢ ¬ ∅ ∈ 𝑆    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))    ⇒   ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Theorem	ndmovordi 7583	Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)
⊢ dom 𝐹 = (𝑆 × 𝑆)    &   ⊢ 𝑅 ⊆ (𝑆 × 𝑆)    &   ⊢ ¬ ∅ ∈ 𝑆    &   ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))    ⇒   ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)
2.3.19.1  Variable-to-class conversion for operations
Theorem	caovclg 7584*	Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
Theorem	caovcld 7585*	Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
Theorem	caovcl 7586*	Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
Theorem	caovcomg 7587*	Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Theorem	caovcomd 7588*	Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Theorem	caovcom 7589*	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)
Theorem	caovassg 7590*	Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Theorem	caovassd 7591*	Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Theorem	caovass 7592*	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))
Theorem	caovcang 7593*	Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
Theorem	caovcand 7594*	Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
Theorem	caovcanrd 7595*	Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    ⇒   ⊢ (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶))
Theorem	caovcan 7596*	Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
⊢ 𝐶 ∈ V    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))
Theorem	caovordig 7597*	Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Theorem	caovordid 7598*	Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Theorem	caovordg 7599*	Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Theorem	caovordd 7600*	Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))