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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuppssfv 6101* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   (𝜑 → (𝐹𝑌) = 𝑍)    &   ((𝜑𝑥𝐷) → 𝐴𝑉)       (𝜑 → ((𝑥𝐷 ↦ (𝐹𝐴)) “ (V ∖ {𝑍})) ⊆ 𝐿)
 
Theoremsuppssov1 6102* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ((𝜑𝑥𝐷) → 𝐴𝑉)    &   ((𝜑𝑥𝐷) → 𝐵𝑅)       (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
 
2.6.13  Function operation
 
Syntaxcof 6103 Extend class notation to include mapping of an operation to a function operation.
class 𝑓 𝑅
 
Syntaxcofr 6104 Extend class notation to include mapping of a binary relation to a function relation.
class 𝑟 𝑅
 
Definitiondf-of 6105* Define the function operation map. The definition is designed so that if 𝑅 is a binary operation, then 𝑓 𝑅 is the analogous operation on functions which corresponds to applying 𝑅 pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
 
Definitiondf-ofr 6106* Define the function relation map. The definition is designed so that if 𝑅 is a binary relation, then 𝑓 𝑅 is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑟 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
 
Theoremofeqd 6107 Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
(𝜑𝑅 = 𝑆)       (𝜑 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
Theoremofeq 6108 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
Theoremofreq 6109 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟 𝑆)
 
Theoremofexg 6110 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
(𝐴𝑉 → ( ∘𝑓 𝑅𝐴) ∈ V)
 
Theoremnfof 6111 Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑥𝑅       𝑥𝑓 𝑅
 
Theoremnfofr 6112 Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑥𝑅       𝑥𝑟 𝑅
 
Theoremoffval 6113* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
 
Theoremofrfval 6114* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)       (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
 
Theoremofvalg 6115 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)    &   ((𝜑𝑋𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)       ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 
Theoremofrval 6116 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)       ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)
 
Theoremofmresval 6117 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐹𝐴)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹𝑓 𝑅𝐺))
 
Theoremoff 6118* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶       (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
 
Theoremoffeq 6119* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶    &   (𝜑𝐻:𝐶𝑈)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐷)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐸)    &   ((𝜑𝑥𝐶) → (𝐷𝑅𝐸) = (𝐻𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)
 
Theoremofres 6120 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶       (𝜑 → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))
 
Theoremoffval2 6121* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 
Theoremofrfval2 6122* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
 
Theoremsuppssof1 6123* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   (𝜑𝐴:𝐷𝑉)    &   (𝜑𝐵:𝐷𝑅)    &   (𝜑𝐷𝑊)       (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
 
Theoremofco 6124 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐻:𝐷𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷𝑋)    &   (𝐴𝐵) = 𝐶       (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
 
Theoremoffveqb 6125* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐻 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)    &   ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)       (𝜑 → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
 
Theoremofc12 6126 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
 
Theoremcaofref 6127* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)       (𝜑𝐹𝑟 𝑅𝐹)
 
Theoremcaofinvl 6128* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝑁:𝑆𝑆)    &   (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))    &   ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)       (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
 
Theoremcaofcom 6129* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
 
Theoremcaofrss 6130* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))       (𝜑 → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
 
Theoremcaoftrn 6131* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))       (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
 
2.6.14  Functions (continued)
 
TheoremresfunexgALT 6132 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5757 but requires ax-pow 4192 and ax-un 4451. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunexg 6133 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunex2g 6134 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((𝐴𝑉 ∧ Fun 𝐵) → (𝐴𝐵) ∈ V)
 
TheoremfnexALT 6135 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5319. This version of fnex 5758 uses ax-pow 4192 and ax-un 4451, whereas fnex 5758 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 
Theoremfunexw 6136 Weak version of funex 5759 that holds without ax-coll 4133. 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 6137* Weak version of mptex 5762 that holds without ax-coll 4133. 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 6138 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 5759. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
 
Theoremfocdmex 6139 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
 
Theoremf1dmex 6140 If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theoremabrexex 6141* Existence of a class abstraction of existentially restricted sets. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be thought of as 𝐵(𝑥). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5761, funex 5759, fnex 5758, resfunexg 5757, and funimaexg 5319. See also abrexex2 6148. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
 
Theoremabrexexg 6142* Existence of a class abstraction of existentially restricted sets. 𝑥 is normally a free-variable parameter in 𝐵. The antecedent assures us that 𝐴 is a set. (Contributed by NM, 3-Nov-2003.)
(𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
 
Theoremiunexg 6143* 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 6144* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
 
Theoremopabex3d 6145* 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)
 
Theoremopabex3 6146* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ V    &   (𝑥𝐴 → {𝑦𝜑} ∈ V)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 
Theoremiunex 6147* 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 6148* Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 6141. (Contributed by NM, 12-Sep-2004.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 
Theoremabexssex 6149* 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 6150* 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
 
Theoremoprabexd 6151* 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 6152* 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 6153* 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 6154* 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 6155* 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 6141. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
 
Theoremab2rexex2 6156* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 6148. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   {𝑧𝜑} ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
 
TheoremxpexgALT 6157 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4758 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
 
Theoremoffval3 6158* 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 6159 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → ((𝐹𝑓 𝑅𝐺) ↾ 𝐷) = ((𝐹𝐷) ∘𝑓 𝑅(𝐺𝐷)))
 
Theoremofmres 6160* 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 6161, allowing it to be used as a function or structure argument. By ofmresval 6117, 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 6161 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)
 
2.6.15  First and second members of an ordered pair
 
Syntaxc1st 6162 Extend the definition of a class to include the first member an ordered pair function.
class 1st
 
Syntaxc2nd 6163 Extend the definition of a class to include the second member an ordered pair function.
class 2nd
 
Definitiondf-1st 6164 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6170 proves that it does this. For example, (1st ‘⟨ 3 , 4 ) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5128 and op1stb 4496). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (𝑥 ∈ V ↦ dom {𝑥})
 
Definitiondf-2nd 6165 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6171 proves that it does this. For example, (2nd ‘⟨ 3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5131 and op2ndb 5130). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (𝑥 ∈ V ↦ ran {𝑥})
 
Theorem1stvalg 6166 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.)
(𝐴 ∈ V → (1st𝐴) = dom {𝐴})
 
Theorem2ndvalg 6167 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.)
(𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
 
Theorem1st0 6168 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ‘∅) = ∅
 
Theorem2nd0 6169 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ‘∅) = ∅
 
Theoremop1st 6170 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
 
Theoremop2nd 6171 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
 
Theoremop1std 6172 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 6173 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 6174 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
 
Theoremop2ndg 6175 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
 
Theoremot1stg 6176 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 6176, ot2ndg 6177, ot3rdgg 6178.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
 
Theoremot2ndg 6177 Extract the second member of an ordered triple. (See ot1stg 6176 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
 
Theoremot3rdgg 6178 Extract the third member of an ordered triple. (See ot1stg 6176 comment.) (Contributed by NM, 3-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
 
Theorem1stval2 6179 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 6180 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})
 
Theoremfo1st 6181 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 6182 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
 
Theoremf1stres 6183 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 6184 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
 
Theoremfo1stresm 6185* Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
 
Theoremfo2ndresm 6186* Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(∃𝑥 𝑥𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
 
Theorem1stcof 6187 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)
 
Theorem2ndcof 6188 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)
 
Theoremxp1st 6189 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
 
Theoremxp2nd 6190 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (2nd𝐴) ∈ 𝐶)
 
Theorem1stexg 6191 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(𝐴𝑉 → (1st𝐴) ∈ V)
 
Theorem2ndexg 6192 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(𝐴𝑉 → (2nd𝐴) ∈ V)
 
Theoremelxp6 6193 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5134. (Contributed by NM, 9-Oct-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
 
Theoremelxp7 6194 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5134. (Contributed by NM, 19-Aug-2006.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
 
Theoremoprssdmm 6195* Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
((𝜑𝑢𝑆) → ∃𝑣 𝑣𝑢)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   (𝜑 → Rel 𝐹)       (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
 
Theoremeqopi 6196 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 6197* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
(𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)}
 
Theoremunielxp 6198 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
 
Theorem1st2nd2 6199 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 = ⟨(1st𝐴), (2nd𝐴)⟩)
 
Theoremxpopth 6200 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
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