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Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelxp5 7701 Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7700 when the double intersection does not create class existence problems (caused by int0 4873). (Contributed by NM, 1-Aug-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
 
Theoremcnvexg 7702 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
(𝐴𝑉𝐴 ∈ V)
 
Theoremcnvex 7703 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
𝐴 ∈ V       𝐴 ∈ V
 
Theoremrelcnvexb 7704 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
(Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
 
Theoremf1oexrnex 7705 If the range of a 1-1 onto function is a set, the function itself is a set. (Contributed by AV, 2-Jun-2019.)
((𝐹:𝐴1-1-onto𝐵𝐵𝑉) → 𝐹 ∈ V)
 
Theoremf1oexbi 7706* There is a one-to-one onto function from a set to a second set iff there is a one-to-one onto function from the second set to the first set. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
(∃𝑓 𝑓:𝐴1-1-onto𝐵 ↔ ∃𝑔 𝑔:𝐵1-1-onto𝐴)
 
Theoremcoexg 7707 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoremcoex 7708 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theoremfuncnvuni 7709* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 6449 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
(∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
 
Theoremfun11uni 7710* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
(∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝐴 ∧ Fun 𝐴))
 
Theoremfex2 7711 A function with bounded domain and range is a set. This version of fex 7042 is proven without the Axiom of Replacement ax-rep 5179, but depends on ax-un 7523, which is not required for the proof of fex 7042. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
 
Theoremfabexg 7712* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
 
Theoremfabex 7713* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       𝐹 ∈ V
 
Theoremf1oabexg 7714* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
 
Theoremfiunlem 7715* Lemma for fiun 7716 and f1iun 7717. Formerly part of f1iun 7717. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)       (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
 
Theoremfiun 7716* The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7717. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
 
Theoremf1iun 7717* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.) (Proof shortened by AV, 5-Nov-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
 
Theoremfviunfun 7718* The function value of an indexed union is the value of one of the indexed functions. (Contributed by AV, 4-Nov-2023.)
𝑈 = 𝑖𝐼 (𝐹𝑖)       ((Fun 𝑈𝐽𝐼𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
 
Theoremffoss 7719* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
𝐹 ∈ V       (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
 
Theoremf11o 7720* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
𝐹 ∈ V       (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))
 
TheoremresfunexgALT 7721 Alternate proof of resfunexg 7031, shorter but requiring ax-pow 5258 and ax-un 7523. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunexg 7722 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunex2g 7723 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((𝐴𝑉 ∧ Fun 𝐵) → (𝐴𝐵) ∈ V)
 
TheoremfnexALT 7724 Alternate proof of fnex 7033, derived using the Axiom of Replacement in the form of funimaexg 6466. This version uses ax-pow 5258 and ax-un 7523, whereas fnex 7033 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 
Theoremfunexw 7725 Weak version of funex 7035 that holds without ax-rep 5179. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)
 
Theoremmptexw 7726* Weak version of mptex 7039 that holds without ax-rep 5179. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝐴 ∈ V    &   𝐶 ∈ V    &   𝑥𝐴 𝐵𝐶       (𝑥𝐴𝐵) ∈ V
 
Theoremfunrnex 7727 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 7035. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
 
Theoremzfrep6 7728* A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 5192 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 5179. (Contributed by NM, 10-Oct-2003.)
(∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
 
Theoremfornex 7729 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
 
Theoremf1dmex 7730 If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 5179. (Contributed by NM, 4-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theoremf1ovv 7731 The range of a 1-1 onto function is a set iff its domain is a set. (Contributed by AV, 21-Mar-2019.)
(𝐹:𝐴1-1-onto𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
 
Theoremfvclex 7732* Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)
𝐹 ∈ V       {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V
 
Theoremfvresex 7733* Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐴 ∈ V       {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)} ∈ V
 
Theoremabrexexg 7734* Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path mptexg 7037, funex 7035, fnex 7033, resfunexg 7031, and funimaexg 6466. See also abrexex2g 7737. There are partial converses under additional conditions, see for instance abnexg 7541. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
 
Theoremabrexex 7735* Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7734. See also abrexex2 7742. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
 
Theoremiunexg 7736* The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵. (Contributed by NM, 23-Mar-2006.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
 
Theoremabrexex2g 7737* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
 
Theoremopabex3d 7738* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 9-Aug-2024.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
 
Theoremopabex3rd 7739* Existence of an ordered pair abstraction if the second components are elements of a set. (Contributed by AV, 17-Sep-2023.) (Revised by AV, 9-Aug-2024.)
(𝜑𝐴𝑉)    &   ((𝜑𝑦𝐴) → {𝑥𝜓} ∈ V)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ∈ V)
 
Theoremopabex3 7740* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ V    &   (𝑥𝐴 → {𝑦𝜑} ∈ V)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 
Theoremiunex 7741* The existence of an indexed union. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be read informally as 𝐵(𝑥). (Contributed by NM, 13-Oct-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝑥𝐴 𝐵 ∈ V
 
Theoremabrexex2 7742* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7735. (Contributed by NM, 12-Sep-2004.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 
Theoremabexssex 7743* Existence of a class abstraction with an existentially quantified expression. Both 𝑥 and 𝑦 can be free in 𝜑. (Contributed by NM, 29-Jul-2006.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
 
Theoremabexex 7744* A condition where a class abstraction continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
𝐴 ∈ V    &   (𝜑𝑥𝐴)    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝜑} ∈ V
 
Theoremf1oweALT 7745* Alternate proof of f1owe 7162, more direct since not using the isomorphism predicate, but requiring ax-un 7523. (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}       (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
 
Theoremwemoiso 7746* Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 9727. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
(𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
 
Theoremwemoiso2 7747* Thus, there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
(𝑆 We 𝐵 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
 
Theoremoprabexd 7748* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by AV, 9-Aug-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → ∃*𝑧𝜓)    &   (𝜑𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)})       (𝜑𝐹 ∈ V)
 
Theoremoprabex 7749* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥𝐴𝑦𝐵) → ∃*𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}       𝐹 ∈ V
 
Theoremoprabex3 7750* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
𝐻 ∈ V    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}       𝐹 ∈ V
 
Theoremoprabrexex2 7751* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
𝐴 ∈ V    &   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} ∈ V
 
Theoremab2rexex 7752* Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 7735. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
 
Theoremab2rexex2 7753* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7742. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   {𝑧𝜑} ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
 
TheoremxpexgALT 7754 Alternate proof of xpexg 7535 requiring Replacement (ax-rep 5179) but not Power Set (ax-pow 5258). (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
 
Theoremoffval3 7755* General value of (𝐹f 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
 
Theoremoffres 7756 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → ((𝐹f 𝑅𝐺) ↾ 𝐷) = ((𝐹𝐷) ∘f 𝑅(𝐺𝐷)))
 
Theoremofmres 7757* Equivalent expressions for a restriction of the function operation map. Unlike f 𝑅 which is a proper class, ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) can be a set by ofmresex 7758, allowing it to be used as a function or structure argument. By ofmresval 7484, the restricted operation map values are the same as the original values, allowing theorems for f 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.)
( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑓f 𝑅𝑔))
 
Theoremofmresex 7758 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)
 
2.4.8  First and second members of an ordered pair
 
Syntaxc1st 7759 Extend the definition of a class to include the first member an ordered pair function.
class 1st
 
Syntaxc2nd 7760 Extend the definition of a class to include the second member an ordered pair function.
class 2nd
 
Definitiondf-1st 7761 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 7769 proves that it does this. For example, (1st ‘⟨3, 4⟩) = 3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 6088 and op1stb 5355). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (𝑥 ∈ V ↦ dom {𝑥})
 
Definitiondf-2nd 7762 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 7770 proves that it does this. For example, (2nd ‘⟨3, 4⟩) = 4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 6091 and op2ndb 6090). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (𝑥 ∈ V ↦ ran {𝑥})
 
Theorem1stval 7763 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(1st𝐴) = dom {𝐴}
 
Theorem2ndval 7764 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(2nd𝐴) = ran {𝐴}
 
Theorem1stnpr 7765 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝐴 ∈ (V × V) → (1st𝐴) = ∅)
 
Theorem2ndnpr 7766 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝐴 ∈ (V × V) → (2nd𝐴) = ∅)
 
Theorem1st0 7767 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ‘∅) = ∅
 
Theorem2nd0 7768 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ‘∅) = ∅
 
Theoremop1st 7769 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
 
Theoremop2nd 7770 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
 
Theoremop1std 7771 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = 𝐴)
 
Theoremop2ndd 7772 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd𝐶) = 𝐵)
 
Theoremop1stg 7773 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
 
Theoremop2ndg 7774 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
 
Theoremot1stg 7775 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 7775, ot2ndg 7776, ot3rdg 7777.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
 
Theoremot2ndg 7776 Extract the second member of an ordered triple. (See ot1stg 7775 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
 
Theoremot3rdg 7777 Extract the third member of an ordered triple. (See ot1stg 7775 comment.) (Contributed by NM, 3-Apr-2015.)
(𝐶𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
 
Theorem1stval2 7778 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) → (1st𝐴) = 𝐴)
 
Theorem2ndval2 7779 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) → (2nd𝐴) = {𝐴})
 
Theoremoteqimp 7780 The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
(𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝐵 ∧ (2nd𝑇) = 𝐶)))
 
Theoremfo1st 7781 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
 
Theoremfo2nd 7782 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
 
Theorembr1steqg 7783 Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
 
Theorembr2ndeqg 7784 Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
 
Theoremf1stres 7785 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
 
Theoremf2ndres 7786 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
 
Theoremfo1stres 7787 Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
(𝐵 ≠ ∅ → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
 
Theoremfo2ndres 7788 Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
(𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
 
Theorem1st2val 7789* Value of an alternate definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}‘𝐴) = (1st𝐴)
 
Theorem2nd2val 7790* Value of an alternate definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}‘𝐴) = (2nd𝐴)
 
Theorem1stcof 7791 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)
 
Theorem2ndcof 7792 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)
 
Theoremxp1st 7793 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
 
Theoremxp2nd 7794 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (2nd𝐴) ∈ 𝐶)
 
Theoremelxp6 7795 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7700. (Contributed by NM, 9-Oct-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
 
Theoremelxp7 7796 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7700. (Contributed by NM, 19-Aug-2006.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
 
Theoremeqopi 7797 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
 
Theoremxp2 7798* Representation of Cartesian product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
(𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)}
 
Theoremunielxp 7799 The membership relation for a Cartesian product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
 
Theorem1st2nd2 7800 Reconstruction of a member of a Cartesian product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
(𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
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