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Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfunopg 5201 A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝑉𝐵𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)

Theoremfunopab 5202* A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)
(Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑)

Theoremfunopabeq 5203* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}

Theoremfunopab4 5204* A class of ordered pairs of values in the form used by df-mpt 4027 is a function. (Contributed by NM, 17-Feb-2013.)
Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}

Theoremfunmpt 5205 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Fun (𝑥𝐴𝐵)

Theoremfunmpt2 5206 Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
𝐹 = (𝑥𝐴𝐵)       Fun 𝐹

Theoremfunco 5207 The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))

Theoremfunres 5208 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
(Fun 𝐹 → Fun (𝐹𝐴))

Theoremfunssres 5209 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)
((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)

Theoremfun2ssres 5210 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))

Theoremfunun 5211 The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)
(((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))

Theoremfuncnvsn 5212 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5215 via cnvsn 5065, but stating it this way allows us to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.)
Fun {⟨𝐴, 𝐵⟩}

Theoremfunsng 5213 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})

Theoremfnsng 5214 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})

Theoremfunsn 5215 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       Fun {⟨𝐴, 𝐵⟩}

Theoremfuninsn 5216 A function based on the singleton of an ordered pair. Unlike funsng 5213, this holds even if 𝐴 or 𝐵 is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.)
Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))

Theoremfunprg 5217 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})

Theoremfuntpg 5218 A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
(((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩})

Theoremfunpr 5219 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})

Theoremfuntp 5220 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V       ((𝐴𝐵𝐴𝐶𝐵𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})

Theoremfnsn 5221 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {⟨𝐴, 𝐵⟩} Fn {𝐴}

Theoremfnprg 5222 Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})

Theoremfntpg 5223 Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
(((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})

Theoremfntp 5224 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V       ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})

Theoremfun0 5225 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
Fun ∅

Theoremfuncnvcnv 5226 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
(Fun 𝐴 → Fun 𝐴)

Theoremfuncnv2 5227* A simpler equivalence for single-rooted (see funcnv 5228). (Contributed by NM, 9-Aug-2004.)
(Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)

Theoremfuncnv 5228* The converse of a class is a function iff the class is single-rooted, which means that for any 𝑦 in the range of 𝐴 there is at most one 𝑥 such that 𝑥𝐴𝑦. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5227 for a simpler version. (Contributed by NM, 13-Aug-2004.)
(Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)

Theoremfuncnv3 5229* A condition showing a class is single-rooted. (See funcnv 5228). (Contributed by NM, 26-May-2006.)
(Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)

Theoremfuncnveq 5230* Another way of expressing that a class is single-rooted. Counterpart to dffun2 5177. (Contributed by Jim Kingdon, 24-Dec-2018.)
(Fun 𝐴 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑧𝐴𝑦) → 𝑥 = 𝑧))

Theoremfun2cnv 5231* The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that 𝐴 is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
(Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)

Theoremsvrelfun 5232 A single-valued relation is a function. (See fun2cnv 5231 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun 𝐴))

Theoremfncnv 5233* Single-rootedness (see funcnv 5228) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)

Theoremfun11 5234* Two ways of stating that 𝐴 is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
((Fun 𝐴 ∧ Fun 𝐴) ↔ ∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))

Theoremfununi 5235* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
(∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)

Theoremfuncnvuni 5236* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5228 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
(∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)

Theoremfun11uni 5237* 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))

Theoremfunin 5238 The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(Fun 𝐹 → Fun (𝐹𝐺))

Theoremfunres11 5239 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
(Fun 𝐹 → Fun (𝐹𝐴))

Theoremfuncnvres 5240 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
(Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))

Theoremcnvresid 5241 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
( I ↾ 𝐴) = ( I ↾ 𝐴)

Theoremfuncnvres2 5242 The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
(Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))

Theoremfunimacnv 5243 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
(Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))

Theoremfunimass1 5244 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))

Theoremfunimass2 5245 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
((Fun 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)

Theoremimadiflem 5246 One direction of imadif 5247. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))

Theoremimadif 5247 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))

Theoremimainlem 5248 One direction of imain 5249. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
(𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))

Theoremimain 5249 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))

Theoremfunimaexglem 5250 Lemma for funimaexg 5251. It constitutes the interesting part of funimaexg 5251, in which 𝐵 ⊆ dom 𝐴. (Contributed by Jim Kingdon, 27-Dec-2018.)
((Fun 𝐴𝐵𝐶𝐵 ⊆ dom 𝐴) → (𝐴𝐵) ∈ V)

Theoremfunimaexg 5251 Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Theoremfunimaex 5252 The image of a set under any function is also a set. Equivalent of Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)
𝐵 ∈ V       (Fun 𝐴 → (𝐴𝐵) ∈ V)

Theoremisarep1 5253* Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
(𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)

Theoremisarep2 5254* Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature "[ i, [ i, i ] => o ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5252. (Contributed by NM, 26-Oct-2006.)
𝐴 ∈ V    &   𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)       𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)

Theoremfneq1 5255 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Theoremfneq2 5256 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Theoremfneq1d 5257 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Theoremfneq2d 5258 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Theoremfneq12d 5259 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))

Theoremfneq12 5260 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹 = 𝐺𝐴 = 𝐵) → (𝐹 Fn 𝐴𝐺 Fn 𝐵))

Theoremfneq1i 5261 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹 Fn 𝐴𝐺 Fn 𝐴)

Theoremfneq2i 5262 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
𝐴 = 𝐵       (𝐹 Fn 𝐴𝐹 Fn 𝐵)

Theoremnffn 5263 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 Fn 𝐴

Theoremfnfun 5264 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Fun 𝐹)

Theoremfnrel 5265 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Rel 𝐹)

Theoremfndm 5266 The domain of a function. (Contributed by NM, 2-Aug-1994.)
(𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)

Theoremfunfni 5267 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)       ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)

Theoremfndmu 5268 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
((𝐹 Fn 𝐴𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Theoremfnbr 5269 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)

Theoremfnop 5270 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵𝐴)

Theoremfneu 5271* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)

Theoremfneu2 5272* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)

Theoremfnun 5273 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))

Theoremfnunsn 5274 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝜑𝑋 ∈ V)    &   (𝜑𝑌 ∈ V)    &   (𝜑𝐹 Fn 𝐷)    &   𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})    &   𝐸 = (𝐷 ∪ {𝑋})    &   (𝜑 → ¬ 𝑋𝐷)       (𝜑𝐺 Fn 𝐸)

Theoremfnco 5275 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)

Theoremfnresdm 5276 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
(𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)

Theoremfnresdisj 5277 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
(𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))

Theorem2elresin 5278 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))

Theoremfnssresb 5279 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
(𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))

Theoremfnssres 5280 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)

Theoremfnresin1 5281 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))

Theoremfnresin2 5282 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))

Theoremfnres 5283* An equivalence for functionality of a restriction. Compare dffun8 5195. (Contributed by Mario Carneiro, 20-May-2015.)
((𝐹𝐴) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)

Theoremfnresi 5284 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
( I ↾ 𝐴) Fn 𝐴

Theoremfnima 5285 The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)

Theoremfn0 5286 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn ∅ ↔ 𝐹 = ∅)

Theoremfnimadisj 5287 A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)

Theoremfnimaeq0 5288 Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))

Theoremdfmpt3 5289 Alternate definition for the maps-to notation df-mpt 4027. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})

Theoremfnopabg 5290* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)

Theoremfnopab 5291* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
(𝑥𝐴 → ∃!𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       𝐹 Fn 𝐴

Theoremmptfng 5292* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)

Theoremfnmpt 5293* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)

Theoremmpt0 5294 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝑥 ∈ ∅ ↦ 𝐴) = ∅

Theoremfnmpti 5295* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V    &   𝐹 = (𝑥𝐴𝐵)       𝐹 Fn 𝐴

Theoremdmmpti 5296* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V    &   𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = 𝐴

Theoremdmmptd 5297* The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)

Theoremmptun 5298 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))

Theoremfeq1 5299 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))

Theoremfeq2 5300 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))

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