P. 68 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	oveq2i 6701	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
Theorem	oveq12i 6702	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴𝐹𝐶) = (𝐵𝐹𝐷)
Theorem	oveqi 6703	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
Theorem	oveq123i 6704	Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    &   ⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
Theorem	oveq1d 6705	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
Theorem	oveq2d 6706	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
Theorem	oveqd 6707	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
Theorem	oveq12d 6708	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveqan12d 6709	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveqan12rd 6710	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
Theorem	oveq123d 6711	Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
Theorem	ovrspc2v 6712*	If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
Theorem	oveqrspc2v 6713*	Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
Theorem	oveqdr 6714	Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
Theorem	nfovd 6715	Deduction version of bound-variable hypothesis builder nfov 6716. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐹)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))
Theorem	nfov 6716	Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴𝐹𝐵)
Theorem	oprabid 6717	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 20-Mar-2013.)
⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
Theorem	ovex 6718	The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)
⊢ (𝐴𝐹𝐵) ∈ V
Theorem	ovexi 6719	The result of an operation is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 = (𝐵𝐹𝐶)    ⇒   ⊢ 𝐴 ∈ V
Theorem	ovexd 6720	The result of an operation is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → (𝐴𝐹𝐵) ∈ V)
Theorem	ovssunirn 6721	The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝑋𝐹𝑌) ⊆ ∪ ran 𝐹
Theorem	0ov 6722	Operation value of the empty set. (Contributed by AV, 15-May-2021.)
⊢ (𝐴∅𝐵) = ∅
Theorem	ovprc 6723	The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
Theorem	ovprc1 6724	The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝐵) = ∅)
Theorem	ovprc2 6725	The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ Rel dom 𝐹    ⇒   ⊢ (¬ 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)
Theorem	ovrcl 6726	Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
⊢ Rel dom 𝐹    ⇒   ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	csbov123 6727	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)
Theorem	csbov 6728*	Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵⦋𝐴 / 𝑥⦌𝐹𝐶)
Theorem	csbov12g 6729*	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))
Theorem	csbov1g 6730*	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹𝐶))
Theorem	csbov2g 6731*	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))
Theorem	rspceov 6732*	A frequently used special case of rspc2ev 3355 for operation values. (Contributed by NM, 21-Mar-2007.)
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦))
Theorem	elovimad 6733	Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))
Theorem	fnbrovb 6734	Value of a binary operation expressed as a binary relation. See also fnbrfvb 6274 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.)
⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ⟨𝐴, 𝐵⟩𝐹𝐶))
Theorem	fnotovb 6735	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6275. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.)
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
Theorem	fnotovbOLD 6736	Old proof of fnotovb 6735 obsolete as of 15-Feb-2022. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
Theorem	opabbrex 6737	A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by BJ/AV, 20-Jun-2019.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)
Theorem	opabresex2d 6738*	Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥(𝑊‘𝐺)𝑦) → 𝜓)    &   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V)
Theorem	fvmptopab 6739*	The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.)
⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)    &   ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)})    ⇒   ⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
Theorem	0neqopab 6740	The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	brabv 6741	If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Theorem	brfvopab 6742	The classes involved in a binary relation of a function value which is an ordered-pair class abstraction are sets. (Contributed by AV, 7-Jan-2021.)
⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {⟨𝑦, 𝑧⟩ ∣ 𝜑})    ⇒   ⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	dfoprab2 6743*	Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Theorem	reloprab 6744*	An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
⊢ Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Theorem	oprabv 6745*	If a pair and a class are in a relationship given by a class abstraction of a collection of nested ordered pairs, the involved classes are sets. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
⊢ (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))
Theorem	nfoprab1 6746	The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Theorem	nfoprab2 6747	The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)
⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Theorem	nfoprab3 6748	The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Theorem	nfoprab 6749*	Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
⊢ Ⅎ𝑤𝜑    ⇒   ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Theorem	oprabbid 6750*	Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Theorem	oprabbidv 6751*	Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Theorem	oprabbii 6752*	Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Theorem	ssoprab2 6753	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 5030. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
Theorem	ssoprab2b 6754	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 5031. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓))
Theorem	eqoprab2b 6755	Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 5034. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓))
Theorem	mpt2eq123 6756*	An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
⊢ ((𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))
Theorem	mpt2eq12 6757*	An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸))
Theorem	mpt2eq123dva 6758*	An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))
Theorem	mpt2eq123dv 6759*	An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))
Theorem	mpt2eq123i 6760	An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)
⊢ 𝐴 = 𝐷    &   ⊢ 𝐵 = 𝐸    &   ⊢ 𝐶 = 𝐹    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)
Theorem	mpt2eq3dva 6761*	Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-2013.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))
Theorem	mpt2eq3ia 6762	An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Theorem	mpt2eq3dv 6763*	An equality deduction for the maps to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)
⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))
Theorem	nfmpt21 6764	Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Theorem	nfmpt22 6765	Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Theorem	nfmpt2 6766*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
⊢ Ⅎ𝑧𝐴    &   ⊢ Ⅎ𝑧𝐵    &   ⊢ Ⅎ𝑧𝐶    ⇒   ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Theorem	mpt20 6767	A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅
Theorem	oprab4 6768*	Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}
Theorem	cbvoprab1 6769*	Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Theorem	cbvoprab2 6770*	Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = 𝑤 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}
Theorem	cbvoprab12 6771*	Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑣𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Theorem	cbvoprab12v 6772*	Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Theorem	cbvoprab3 6773*	Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Theorem	cbvoprab3v 6774*	Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Theorem	cbvmpt2x 6775*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6776 allows 𝐵 to be a function of 𝑥. (Contributed by NM, 29-Dec-2014.)
⊢ Ⅎ𝑧𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑤𝐶    &   ⊢ Ⅎ𝑥𝐸    &   ⊢ Ⅎ𝑦𝐸    &   ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐷)    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐸)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸)
Theorem	cbvmpt2 6776*	Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑤𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ Ⅎ𝑦𝐷    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)
Theorem	cbvmpt2v 6777*	Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4782, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)    &   ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)
Theorem	elimdelov 6778	Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). (Contributed by Paul Chapman, 25-Mar-2008.)
⊢ (𝜑 → 𝐶 ∈ (𝐴𝐹𝐵))    &   ⊢ 𝑍 ∈ (𝑋𝐹𝑌)    ⇒   ⊢ if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))
Theorem	ovif 6779	Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶))
Theorem	ovif2 6780	Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 1-Oct-2018.)
⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))
Theorem	ovif12 6781	Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))
Theorem	ifov 6782	Move a conditional outside of an operation. (Contributed by AV, 11-Nov-2019.)
⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵))
Theorem	dmoprab 6783*	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}
Theorem	dmoprabss 6784*	The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Theorem	rnoprab 6785*	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 ∣ ∃𝑥∃𝑦𝜑}
Theorem	rnoprab2 6786*	The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑}
Theorem	reldmoprab 6787*	The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Theorem	oprabss 6788*	Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)
Theorem	eloprabga 6789*	The law of concretion for operation class abstraction. Compare elopab 5012. (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 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
Theorem	eloprabg 6790*	The law of concretion for operation class abstraction. Compare elopab 5012. (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 6791*	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	mpt2v 6792*	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	mpt2mptx 6793*	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	mpt2mpt 6794*	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	mpt2difsnif 6795	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	mpt2snif 6796	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	fconstmpt2 6797*	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 6798*	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 6799*	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	resmpt2 6800*	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))