P. 64 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	op1st 6301	Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Theorem	op2nd 6302	Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
Theorem	op1std 6303	Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (1st ‘𝐶) = 𝐴)
Theorem	op2ndd 6304	Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝐶) = 𝐵)
Theorem	op1stg 6305	Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
Theorem	op2ndg 6306	Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
Theorem	ot1stg 6307	Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6307, ot2ndg 6308, ot3rdgg 6309.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
Theorem	ot2ndg 6308	Extract the second member of an ordered triple. (See ot1stg 6307 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
Theorem	ot3rdgg 6309	Extract the third member of an ordered triple. (See ot1stg 6307 comment.) (Contributed by NM, 3-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
Theorem	1stval2 6310	Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴)
Theorem	2ndval2 6311	Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
Theorem	fo1st 6312	The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ 1st :V–onto→V
Theorem	fo2nd 6313	The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ 2nd :V–onto→V
Theorem	f1stres 6314	Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
Theorem	f2ndres 6315	Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
Theorem	fo1stresm 6316*	Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)
Theorem	fo2ndresm 6317*	Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)
Theorem	1stcof 6318	Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵)
Theorem	2ndcof 6319	Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶)
Theorem	xp1st 6320	Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵)
Theorem	xp2nd 6321	Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶)
Theorem	1stexg 6322	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V)
Theorem	2ndexg 6323	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)
Theorem	elxp6 6324	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5219. (Contributed by NM, 9-Oct-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
Theorem	elxp7 6325	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5219. (Contributed by NM, 19-Aug-2006.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
Theorem	oprssdmm 6326*	Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆) → ∃𝑣 𝑣 ∈ 𝑢)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → Rel 𝐹)    ⇒   ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
Theorem	eqopi 6327	Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
Theorem	xp2 6328*	Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)}
Theorem	unielxp 6329	The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
Theorem	1st2nd2 6330	Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
Theorem	xpopth 6331	An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵))
Theorem	eqop 6332	Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
Theorem	eqop2 6333	Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
Theorem	op1steq 6334*	Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
Theorem	2nd1st 6335	Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)
Theorem	1st2nd 6336	Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
Theorem	1stdm 6337	The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅)
Theorem	2ndrn 6338	The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅)
Theorem	1st2ndbr 6339	Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴))
Theorem	releldm2 6340*	Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))
Theorem	reldm 6341*	An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))
Theorem	sbcopeq1a 6342	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3038 that avoids the existential quantifiers of copsexg 4331). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))
Theorem	csbopeq1a 6343	Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analog of csbeq1a 3133). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)
Theorem	dfopab2 6344*	A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}
Theorem	dfoprab3s 6345*	A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}
Theorem	dfoprab3 6346*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Theorem	dfoprab4 6347*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
Theorem	dfoprab4f 6348*	Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
Theorem	opabex2 6349*	Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Theorem	opabn1stprc 6350*	An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.)
⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)
Theorem	dfxp3 6351*	Define the cross product of three classes. Compare df-xp 4726. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
⊢ ((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)}
Theorem	elopabi 6352*	A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Theorem	eloprabi 6353*	A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
Theorem	mpomptsx 6354*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
Theorem	mpompts 6355*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
Theorem	dmmpossx 6356*	The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
Theorem	fmpox 6357*	Functionality, domain and codomain of a class given by the maps-to notation, where 𝐵(𝑥) is not constant but depends on 𝑥. (Contributed by NM, 29-Dec-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷)
Theorem	fmpo 6358*	Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷)
Theorem	fnmpo 6359*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵))
Theorem	fnmpoi 6360*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ 𝐹 Fn (𝐴 × 𝐵)
Theorem	dmmpo 6361*	Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ dom 𝐹 = (𝐴 × 𝐵)
Theorem	mpofvex 6362*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)
Theorem	mpofvexi 6363*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ (𝑅𝐹𝑆) ∈ V
Theorem	ovmpoelrn 6364*	An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
Theorem	dmmpoga 6365*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6361. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Theorem	dmmpog 6366*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6361. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Theorem	mpoexxg 6367*	Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	mpoexg 6368*	Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	mpoexga 6369*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
Theorem	mpoexw 6370*	Weak version of mpoex 6371 that holds without ax-coll 4199. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
Theorem	mpoex 6371*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
Theorem	fnmpoovd 6372*	A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))    &   ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))
Theorem	fmpoco 6373*	Composition of two functions. Variation of fmptco 5806 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))    &   ⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
Theorem	oprabco 6374*	Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶))    ⇒   ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))
Theorem	oprab2co 6375*	Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨𝐶, 𝐷⟩)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷))    ⇒   ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))
Theorem	df1st2 6376*	An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
Theorem	df2nd2 6377*	An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
Theorem	1stconst 6378	The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)
Theorem	2ndconst 6379	The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)
Theorem	dfmpo 6380*	Alternate definition for the maps-to notation df-mpo 6015 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Theorem	cnvf1olem 6381	Lemma for cnvf1o 6382. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ◡{𝐵})) → (𝐶 ∈ ◡𝐴 ∧ 𝐵 = ∪ ◡{𝐶}))
Theorem	cnvf1o 6382*	Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴)
Theorem	f2ndf 6383	The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
Theorem	fo2ndf 6384	The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
Theorem	f1o2ndf1 6385	The 2nd (second component of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
Theorem	algrflem 6386	Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵)
Theorem	algrflemg 6387	Lemma for algrf 12588 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵))
Theorem	xporderlem 6388*	Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
Theorem	poxp 6389*	A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 Po 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐴 × 𝐵))
Theorem	spc2ed 6390*	Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝜒 → ∃𝑥∃𝑦𝜓))
Theorem	cnvoprab 6391*	The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))    &   ⊢ (𝜓 → 𝑎 ∈ (V × V))    ⇒   ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Theorem	f1od2 6392*	Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))    &   ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))    ⇒   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)
Theorem	disjxp1 6393*	The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
Theorem	disjsnxp 6394*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
2.6.16  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 6015) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 6091, ovmpox 6142 and fmpox 6357). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.
Theorem	opeliunxp2f 6395*	Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 4865. (Contributed by AV, 25-Oct-2020.)
⊢ Ⅎ𝑥𝐸    &   ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)    ⇒   ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
Theorem	mpoxopn0yelv 6396*	If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
Theorem	mpoxopoveq 6397*	Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})    ⇒   ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Theorem	mpoxopovel 6398*	Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
Theorem	rbropapd 6399*	Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})    &   ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))))
Theorem	rbropap 6400*	Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.)
⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})    &   ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))