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Theorem List for Intuitionistic Logic Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelovmpt2 5701* Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)    &   𝐶 ∈ V    &   ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)       (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))

Theoremf1ocnvd 5702* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝑊)    &   ((𝜑𝑦𝐵) → 𝐷𝑋)    &   (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))

Theoremf1od 5703* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝑊)    &   ((𝜑𝑦𝐵) → 𝐷𝑋)    &   (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))       (𝜑𝐹:𝐴1-1-onto𝐵)

Theoremf1ocnv2d 5704* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))

Theoremf1o2d 5705* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑𝐹:𝐴1-1-onto𝐵)

Theoremf1opw2 5706* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5707 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
(𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑 → (𝐹𝑎) ∈ V)    &   (𝜑 → (𝐹𝑏) ∈ V)       (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)

Theoremf1opw 5707* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
(𝐹:𝐴1-1-onto𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)

Theoremsuppssfv 5708* 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 5709* 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)

2.6.12  Function operation

Syntaxcof 5710 Extend class notation to include mapping of an operation to a function operation.
class 𝑓 𝑅

Syntaxcofr 5711 Extend class notation to include mapping of a binary relation to a function relation.
class 𝑟 𝑅

Definitiondf-of 5712* 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 5713* 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)}

Theoremofeq 5714 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Theoremofreq 5715 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟 𝑆)

Theoremofexg 5716 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
(𝐴𝑉 → ( ∘𝑓 𝑅𝐴) ∈ V)

Theoremnfof 5717* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑥𝑅       𝑥𝑓 𝑅

Theoremnfofr 5718* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑥𝑅       𝑥𝑟 𝑅

Theoremoffval 5719* 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 5720* 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))

Theoremfnofval 5721 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)    &   (𝜑𝑅 Fn (𝑈 × 𝑉))    &   (𝜑𝐶𝑈)    &   (𝜑𝐷𝑉)       ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Theoremofrval 5722 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 5723 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 5724* 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)

Theoremofres 5725 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 5726* 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 5727* 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 5728* 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 5729 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 5730* 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 5731 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 5732* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)       (𝜑𝐹𝑟 𝑅𝐹)

Theoremcaofinvl 5733* 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 5734* 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 5735* 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 5736* 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))

2.6.13  Functions (continued)

TheoremresfunexgALT 5737 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 5382 but requires ax-pow 3927 and ax-un 4170. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Theoremcofunexg 5738 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Theoremcofunex2g 5739 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((𝐴𝑉 ∧ Fun 𝐵) → (𝐴𝐵) ∈ V)

TheoremfnexALT 5740 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 4983. This version of fnex 5383 uses ax-pow 3927 and ax-un 4170, whereas fnex 5383 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Theoremfunrnex 5741 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 5384. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))

Theoremfornex 5742 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))

Theoremf1dmex 5743 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 5744* 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 5386, funex 5384, fnex 5383, resfunexg 5382, and funimaexg 4983. See also abrexex2 5751. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V

Theoremabrexexg 5745* 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 5746* 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 5747* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)

Theoremopabex3d 5748* 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 5749* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ V    &   (𝑥𝐴 → {𝑦𝜑} ∈ V)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V

Theoremiunex 5750* 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 5751* Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 5744. (Contributed by NM, 12-Sep-2004.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V

Theoremabexssex 5752* 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 5753* 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 5754* 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 5755* 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 5756* 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 5757* 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 5758* 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 5744. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V

Theoremab2rexex2 5759* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 5751. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   {𝑧𝜑} ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V

TheoremxpexgALT 5760 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4452 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 5761* 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 5762 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → ((𝐹𝑓 𝑅𝐺) ↾ 𝐷) = ((𝐹𝐷) ∘𝑓 𝑅(𝐺𝐷)))

Theoremofmres 5763* 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 5764, allowing it to be used as a function or structure argument. By ofmresval 5723, 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 5764 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)

2.6.14  First and second members of an ordered pair

Syntaxc1st 5765 Extend the definition of a class to include the first member an ordered pair function.
class 1st

Syntaxc2nd 5766 Extend the definition of a class to include the second member an ordered pair function.
class 2nd

Definitiondf-1st 5767 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 5773 proves that it does this. For example, (1st ‘⟨ 3 , 4 ) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 4802 and op1stb 4209). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (𝑥 ∈ V ↦ dom {𝑥})

Definitiondf-2nd 5768 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5774 proves that it does this. For example, (2nd ‘⟨ 3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4805 and op2ndb 4804). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (𝑥 ∈ V ↦ ran {𝑥})

Theorem1stvalg 5769 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 5770 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 5771 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ‘∅) = ∅

Theorem2nd0 5772 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ‘∅) = ∅

Theoremop1st 5773 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (1st ‘⟨𝐴, 𝐵⟩) = 𝐴

Theoremop2nd 5774 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Theoremop1std 5775 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 5776 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 5777 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Theoremop2ndg 5778 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Theoremot1stg 5779 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 5779, ot2ndg 5780, ot3rdgg 5781.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)

Theoremot2ndg 5780 Extract the second member of an ordered triple. (See ot1stg 5779 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)

Theoremot3rdgg 5781 Extract the third member of an ordered triple. (See ot1stg 5779 comment.) (Contributed by NM, 3-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)

Theorem1stval2 5782 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 5783 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 5784 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 5785 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 5786 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 5787 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 5788* 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 5789* 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 5790 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)

Theorem2ndcof 5791 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)

Theoremxp1st 5792 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)

Theoremxp2nd 5793 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (2nd𝐴) ∈ 𝐶)

Theorem1stexg 5794 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(𝐴𝑉 → (1st𝐴) ∈ V)

Theorem2ndexg 5795 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(𝐴𝑉 → (2nd𝐴) ∈ V)

Theoremelxp6 5796 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4808. (Contributed by NM, 9-Oct-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Theoremelxp7 5797 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4808. (Contributed by NM, 19-Aug-2006.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Theoremeqopi 5798 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 5799* 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 5800 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))

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