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Theorem List for Metamath Proof Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
2.4.7  Relations and functions (cont.)
 
Theoremdmexg 7601 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → dom 𝐴 ∈ V)
 
Theoremrnexg 7602 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
(𝐴𝑉 → ran 𝐴 ∈ V)
 
Theoremdmexd 7603 The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → dom 𝐴 ∈ V)
 
Theoremdmex 7604 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
𝐴 ∈ V       dom 𝐴 ∈ V
 
Theoremrnex 7605 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
𝐴 ∈ V       ran 𝐴 ∈ V
 
Theoremiprc 7606 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 21795. (Contributed by NM, 1-Jan-2007.)
¬ I ∈ V
 
Theoremresiexg 7607 The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6970). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
(𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
 
Theoremimaexg 7608 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theoremimaex 7609 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoremexse2 7610 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅𝑉𝑅 Se 𝐴)
 
Theoremxpexr 7611 If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)
((𝐴 × 𝐵) ∈ 𝐶 → (𝐴 ∈ V ∨ 𝐵 ∈ V))
 
Theoremxpexr2 7612 If a nonempty Cartesian product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)
(((𝐴 × 𝐵) ∈ 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theoremxpexcnv 7613 A condition where the converse of xpex 7464 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)
 
Theoremsoex 7614 If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)
((𝑅 Or 𝐴𝑅𝑉) → 𝐴 ∈ V)
 
Theoremelxp4 7615 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7616, elxp6 7714, and elxp7 7715. (Contributed by NM, 17-Feb-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
 
Theoremelxp5 7616 Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7615 when the double intersection does not create class existence problems (caused by int0 4883). (Contributed by NM, 1-Aug-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
 
Theoremcnvexg 7617 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 7618 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 7619 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
(Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
 
Theoremf1oexrnex 7620 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 7621* 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 7622 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoremcoex 7623 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theoremfuncnvuni 7624* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 6417 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
(∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
 
Theoremfun11uni 7625* 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 7626 A function with bounded domain and range is a set. This version of fex 6981 is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
 
Theoremfabexg 7627* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
 
Theoremfabex 7628* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       𝐹 ∈ V
 
Theoremdmfex 7629 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
 
Theoremf1oabexg 7630* 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)
 
Theoremfiunlemw 7631* Version of fiunlem 7634 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)       (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
 
Theoremfiunw 7632* Version of fiun 7635 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
 
Theoremf1iunw 7633* Version of f1iun 7636 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
 
Theoremfiunlem 7634* Lemma for fiun 7635 and f1iun 7636. Formerly part of f1iun 7636. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)       (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
 
Theoremfiun 7635* The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7636. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
 
Theoremf1iun 7636* 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 7637* 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 7638* 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 7639* 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 7640 Alternate proof of resfunexg 6970, shorter but requiring ax-pow 5258 and ax-un 7450. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunexg 7641 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunex2g 7642 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((𝐴𝑉 ∧ Fun 𝐵) → (𝐴𝐵) ∈ V)
 
TheoremfnexALT 7643 Alternate proof of fnex 6972, derived using the Axiom of Replacement in the form of funimaexg 6434. This version uses ax-pow 5258 and ax-un 7450, whereas fnex 6972 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 
Theoremfunexw 7644 Weak version of funex 6974 that holds without ax-rep 5182. 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 7645* Weak version of mptex 6978 that holds without ax-rep 5182. 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 7646 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 6974. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
 
Theoremzfrep6 7647* 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 5195 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 5182. (Contributed by NM, 10-Oct-2003.)
(∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
 
Theoremfornex 7648 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
 
Theoremf1dmex 7649 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 5182. (Contributed by NM, 4-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theoremf1ovv 7650 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 7651* Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)
𝐹 ∈ V       {𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ∈ V
 
Theoremfvresex 7652* 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 7653* 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 6976, funex 6974, fnex 6972, resfunexg 6970, and funimaexg 6434. See also abrexex2g 7656. There are partial converses under additional conditions, see for instance abnexg 7466. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
 
Theoremabrexex 7654* Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7653. See also abrexex2 7661. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
 
Theoremiunexg 7655* 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 7656* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
 
Theoremopabex3d 7657* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
(𝜑𝐴 ∈ V)    &   ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
 
Theoremopabex3rd 7658* Existence of an ordered pair abstraction if the second components are elements of a set. (Contributed by AV, 17-Sep-2023.)
(𝜑𝐴 ∈ V)    &   ((𝜑𝑦𝐴) → {𝑥𝜓} ∈ V)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ∈ V)
 
Theoremopabex3 7659* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ V    &   (𝑥𝐴 → {𝑦𝜑} ∈ V)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 
Theoremiunex 7660* 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 7661* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7654. (Contributed by NM, 12-Sep-2004.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 
Theoremabexssex 7662* 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 7663* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
𝐴 ∈ V    &   (𝜑𝑥𝐴)    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝜑} ∈ V
 
Theoremf1oweALT 7664* Alternate proof of f1owe 7095, more direct since not using the isomorphism predicate, but requiring ax-un 7450. (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 7665* Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 9528. (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 7666* 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 7667* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → ∃*𝑧𝜓)    &   (𝜑𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)})       (𝜑𝐹 ∈ V)
 
Theoremoprabex 7668* 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 7669* 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 7670* 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 7671* 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 7654. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
 
Theoremab2rexex2 7672* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7661. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   {𝑧𝜑} ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
 
TheoremxpexgALT 7673 Alternate proof of xpexg 7461 requiring Replacement (ax-rep 5182) 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 7674* 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 7675 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → ((𝐹f 𝑅𝐺) ↾ 𝐷) = ((𝐹𝐷) ∘f 𝑅(𝐺𝐷)))
 
Theoremofmres 7676* 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 7677, allowing it to be used as a function or structure argument. By ofmresval 7411, 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 7677 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 7678 Extend the definition of a class to include the first member an ordered pair function.
class 1st
 
Syntaxc2nd 7679 Extend the definition of a class to include the second member an ordered pair function.
class 2nd
 
Definitiondf-1st 7680 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 7688 proves that it does this. For example, (1st ‘⟨3, 4⟩) = 3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 6076 and op1stb 5355). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (𝑥 ∈ V ↦ dom {𝑥})
 
Definitiondf-2nd 7681 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 7689 proves that it does this. For example, (2nd ‘⟨3, 4⟩) = 4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 6079 and op2ndb 6078). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (𝑥 ∈ V ↦ ran {𝑥})
 
Theorem1stval 7682 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 7683 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 7684 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝐴 ∈ (V × V) → (1st𝐴) = ∅)
 
Theorem2ndnpr 7685 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝐴 ∈ (V × V) → (2nd𝐴) = ∅)
 
Theorem1st0 7686 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ‘∅) = ∅
 
Theorem2nd0 7687 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ‘∅) = ∅
 
Theoremop1st 7688 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
 
Theoremop2nd 7689 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
 
Theoremop1std 7690 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 7691 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 7692 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
 
Theoremop2ndg 7693 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
 
Theoremot1stg 7694 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 7694, ot2ndg 7695, ot3rdg 7696.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
 
Theoremot2ndg 7695 Extract the second member of an ordered triple. (See ot1stg 7694 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
 
Theoremot3rdg 7696 Extract the third member of an ordered triple. (See ot1stg 7694 comment.) (Contributed by NM, 3-Apr-2015.)
(𝐶𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
 
Theorem1stval2 7697 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 7698 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 7699 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 7700 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
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