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Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremforn 5201 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
 
Theoremdffo2 5202 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
 
Theoremfoima 5203 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
 
Theoremdffn4 5204 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
 
Theoremfunforn 5205 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun 𝐴𝐴:dom 𝐴onto→ran 𝐴)
 
Theoremfodmrnu 5206 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremfores 5207 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
 
Theoremfoco 5208 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
 
Theoremf1oeq1 5209 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
 
Theoremf1oeq2 5210 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
 
Theoremf1oeq3 5211 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
 
Theoremf1oeq23 5212 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
 
Theoremf1eq123d 5213 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐵1-1𝐷))
 
Theoremfoeq123d 5214 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
 
Theoremf1oeq123d 5215 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
 
Theoremnff1o 5216 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴1-1-onto𝐵
 
Theoremf1of1 5217 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
 
Theoremf1of 5218 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
 
Theoremf1ofn 5219 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐴)
 
Theoremf1ofun 5220 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
 
Theoremf1orel 5221 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
 
Theoremf1odm 5222 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
 
Theoremdff1o2 5223 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
 
Theoremdff1o3 5224 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
 
Theoremf1ofo 5225 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
 
Theoremdff1o4 5226 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
 
Theoremdff1o5 5227 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
 
Theoremf1orn 5228 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
(𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
 
Theoremf1f1orn 5229 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
(𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
 
Theoremf1oabexg 5230* 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)
 
Theoremf1ocnv 5231 The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
 
Theoremf1ocnvb 5232 A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
(Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))
 
Theoremf1ores 5233 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
 
Theoremf1orescnv 5234 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
((Fun 𝐹 ∧ (𝐹𝑅):𝑅1-1-onto𝑃) → (𝐹𝑃):𝑃1-1-onto𝑅)
 
Theoremf1imacnv 5235 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoimacnv 5236 A reverse version of f1imacnv 5235. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoun 5237 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
(((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))
 
Theoremf1oun 5238 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
(((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
 
Theoremfun11iun 5239* 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.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
 
Theoremresdif 5240 The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
 
Theoremf1oco 5241 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
((𝐹:𝐵1-1-onto𝐶𝐺:𝐴1-1-onto𝐵) → (𝐹𝐺):𝐴1-1-onto𝐶)
 
Theoremf1cnv 5242 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
 
Theoremfuncocnv2 5243 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremfococnv2 5244 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
(𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
 
Theoremf1ococnv2 5245 The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
(𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
 
Theoremf1cocnv2 5246 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremf1ococnv1 5247 The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
 
Theoremf1cocnv1 5248 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
 
Theoremfuncoeqres 5249 Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((Fun 𝐺 ∧ (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))
 
Theoremffoss 5250* 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 5251* 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))
 
Theoremf10 5252 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
∅:∅–1-1𝐴
 
Theoremf1o00 5253 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
(𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremfo00 5254 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
(𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremf1o0 5255 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
∅:∅–1-1-onto→∅
 
Theoremf1oi 5256 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
( I ↾ 𝐴):𝐴1-1-onto𝐴
 
Theoremf1ovi 5257 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
I :V–1-1-onto→V
 
Theoremf1osn 5258 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
 
Theoremf1osng 5259 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
 
Theoremf1oprg 5260 An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷}))
 
Theoremtz6.12-2 5261* Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
 
Theoremfveu 5262* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
 
Theorembrprcneu 5263* If 𝐴 is a proper class and 𝐹 is any class, then there is no unique set which is related to 𝐴 through the binary relation 𝐹. (Contributed by Scott Fenton, 7-Oct-2017.)
𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
 
Theoremfvprc 5264 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
𝐴 ∈ V → (𝐹𝐴) = ∅)
 
Theoremfv2 5265* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹𝐴) = {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥)}
 
Theoremdffv3g 5266* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
(𝐴𝑉 → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
 
Theoremdffv4g 5267* The previous definition of function value, from before the operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4770), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
(𝐴𝑉 → (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}})
 
Theoremelfv 5268* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
(𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
 
Theoremfveq1 5269 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfveq2 5270 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
 
Theoremfveq1i 5271 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
𝐹 = 𝐺       (𝐹𝐴) = (𝐺𝐴)
 
Theoremfveq1d 5272 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfveq2i 5273 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
𝐴 = 𝐵       (𝐹𝐴) = (𝐹𝐵)
 
Theoremfveq2d 5274 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐹𝐵))
 
Theorem2fveq3 5275 Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
(𝐴 = 𝐵 → (𝐹‘(𝐺𝐴)) = (𝐹‘(𝐺𝐵)))
 
Theoremfveq12i 5276 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
𝐹 = 𝐺    &   𝐴 = 𝐵       (𝐹𝐴) = (𝐺𝐵)
 
Theoremfveq12d 5277 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐺𝐵))
 
Theoremfveqeq2d 5278 Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
(𝜑𝐴 = 𝐵)       (𝜑 → ((𝐹𝐴) = 𝐶 ↔ (𝐹𝐵) = 𝐶))
 
Theoremfveqeq2 5279 Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
(𝐴 = 𝐵 → ((𝐹𝐴) = 𝐶 ↔ (𝐹𝐵) = 𝐶))
 
Theoremnffv 5280 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹𝐴)
 
Theoremnffvmpt1 5281* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑥((𝑥𝐴𝐵)‘𝐶)
 
Theoremnffvd 5282 Deduction version of bound-variable hypothesis builder nffv 5280. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐹)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝐹𝐴))
 
Theoremfunfveu 5283* A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)
 
Theoremfvss 5284* The value of a function is a subset of 𝐵 if every element that could be a candidate for the value is a subset of 𝐵. (Contributed by Mario Carneiro, 24-May-2019.)
(∀𝑥(𝐴𝐹𝑥𝑥𝐵) → (𝐹𝐴) ⊆ 𝐵)
 
Theoremfvssunirng 5285 The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
(𝐴 ∈ V → (𝐹𝐴) ⊆ ran 𝐹)
 
Theoremrelfvssunirn 5286 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
(Rel 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
 
Theoremfunfvex 5287 The value of a function exists. A special case of Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ V)
 
Theoremrelrnfvex 5288 If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
 
Theoremfvexg 5289 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
((𝐹𝑉𝐴𝑊) → (𝐹𝐴) ∈ V)
 
Theoremfvex 5290 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
𝐹𝑉    &   𝐴𝑊       (𝐹𝐴) ∈ V
 
Theoremsefvex 5291 If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
((𝐹 Se V ∧ 𝐴 ∈ V) → (𝐹𝐴) ∈ V)
 
Theoremfvifdc 5292 Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)
(DECID 𝜑 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵)))
 
Theoremfv3 5293* Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹𝐴) = {𝑥 ∣ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}
 
Theoremfvres 5294 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
 
Theoremfunssfv 5295 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
 
Theoremtz6.12-1 5296* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
 
Theoremtz6.12 5297* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
 
Theoremtz6.12f 5298* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
𝑦𝐹       ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
 
Theoremtz6.12c 5299* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
 
Theoremndmfvg 5300 The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)
((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) = ∅)
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