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

Theoremfnima 5201 The image of a function's domain is its range. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 4-Nov-2004.) (Revised by set.mm contributors, 18-Sep-2011.)
(F Fn A → (FA) = ran F)

Theoremfn0 5202 A function with empty domain is empty. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 15-Apr-1998.) (Revised by set.mm contributors, 18-Sep-2011.)
(F Fn F = )

Theoremfnimadisj 5203 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) = )

Theoremiunfopab 5204* Two ways to express a function as a class of ordered pairs. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Sep-2011.) (Contributed by set.mm contributors, 19-Dec-2008.)
B V       x A {x, B} = {x, y (x A y = B)}

Theoremfnopabg 5205* 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 (x A φ)}       (x A ∃!yφF Fn A)

Theoremfnopab2g 5206* Functionality and domain of an ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Mar-2006.)
F = {x, y (x A y = B)}       (x A B V ↔ F Fn A)

Theoremfnopab 5207* Functionality and domain of an ordered-pair class abstraction. (Contributed by set.mm contributors, 5-Mar-1996.)
(x A∃!yφ)    &   F = {x, y (x A φ)}       F Fn A

Theoremfnopab2 5208* Functionality and domain of an ordered-pair class abstraction. (Contributed by set.mm contributors, 29-Jan-2004.)
B V    &   F = {x, y (x A y = B)}       F Fn A

Theoremdmopab2 5209* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by set.mm contributors, 6-Sep-2005.)
B V    &   F = {x, y (x A y = B)}       dom F = A

Theoremfeq1 5210 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(F = G → (F:A–→BG:A–→B))

Theoremfeq2 5211 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(A = B → (F:A–→CF:B–→C))

Theoremfeq3 5212 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(A = B → (F:C–→AF:C–→B))

Theoremfeq23 5213 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (The proof was shortened by Andrew Salmon, 17-Sep-2011.)
((A = C B = D) → (F:A–→BF:C–→D))

Theoremfeq1d 5214 Equality deduction for functions. (Contributed by set.mm contributors, 19-Feb-2008.)
(φF = G)       (φ → (F:A–→BG:A–→B))

Theoremfeq2d 5215 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (F:A–→CF:B–→C))

Theoremfeq12d 5216 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φF = G)    &   (φA = B)       (φ → (F:A–→CG:B–→C))

Theoremfeq1i 5217 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
F = G       (F:A–→BG:A–→B)

Theoremfeq2i 5218 Equality inference for functions. (Contributed by set.mm contributors, 5-Sep-2011.)
A = B       (F:A–→CF:B–→C)

Theoremfeq23i 5219 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
A = C    &   B = D       (F:A–→BF:C–→D)

Theoremfeq23d 5220 Equality deduction for functions. (Contributed by set.mm contributors, 8-Jun-2013.)
(φA = C)    &   (φB = D)       (φ → (F:A–→BF:C–→D))

Theoremnff 5221 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xF    &   xA    &   xB       x F:A–→B

Theoremelimf 5222 Eliminate a mapping hypothesis for the weak deduction theorem dedth 3703, when a special case G:A–→B is provable, in order to convert F:A–→B from a hypothesis to an antecedent. (Contributed by set.mm contributors, 24-Aug-2006.)
G:A–→B        if(F:A–→B, F, G):A–→B

Theoremffn 5223 A mapping is a function. (Contributed by set.mm contributors, 2-Aug-1994.)
(F:A–→BF Fn A)

Theoremdffn2 5224 Any function is a mapping into V. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 31-Oct-1995.) (Revised by set.mm contributors, 18-Sep-2011.)
(F Fn AF:A–→V)

Theoremffun 5225 A mapping is a function. (Contributed by set.mm contributors, 3-Aug-1994.)
(F:A–→B → Fun F)

Theoremfdm 5226 The domain of a mapping. (Contributed by set.mm contributors, 2-Aug-1994.)
(F:A–→B → dom F = A)

Theoremfdmi 5227 The domain of a mapping. (Contributed by set.mm contributors, 28-Jul-2008.)
F:A–→B       dom F = A

Theoremfrn 5228 The range of a mapping. (Contributed by set.mm contributors, 3-Aug-1994.)
(F:A–→B → ran F B)

Theoremdffn3 5229 A function maps to its range. (Contributed by set.mm contributors, 1-Sep-1999.)
(F Fn AF:A–→ran F)

Theoremfss 5230 Expanding the codomain of a mapping. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 10-May-1998.) (Revised by set.mm contributors, 18-Sep-2011.)
((F:A–→B B C) → F:A–→C)

Theoremfco 5231 Composition of two mappings. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 29-Aug-1999.) (Revised by set.mm contributors, 18-Sep-2011.)
((F:B–→C G:A–→B) → (F G):A–→C)

Theoremfssxp 5232 A mapping is a class of ordered pairs. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 18-Sep-2011.)
(F:A–→BF (A × B))

Theoremfunssxp 5233 Two ways of specifying a partial function from A to B. (Contributed by set.mm contributors, 13-Nov-2007.)
((Fun F F (A × B)) ↔ (F:dom F–→B dom F A))

Theoremffdm 5234 A mapping is a partial function. (Contributed by set.mm contributors, 25-Nov-2007.)
(F:A–→B → (F:dom F–→B dom F A))

Theoremopelf 5235 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by set.mm contributors, 9-Jan-2015.)
((F:A–→B C, D F) → (C A D B))

Theoremfun 5236 The union of two functions with disjoint domains. (Contributed by set.mm contributors, 22-Sep-2004.)
(((F:A–→C G:B–→D) (AB) = ) → (FG):(AB)–→(CD))

Theoremfnfco 5237 Composition of two functions. (Contributed by set.mm contributors, 22-May-2006.)
((F Fn A G:B–→A) → (F G) Fn B)

Theoremfssres 5238 Restriction of a function with a subclass of its domain. (Contributed by set.mm contributors, 23-Sep-2004.)
((F:A–→B C A) → (F C):C–→B)

Theoremfssres2 5239 Restriction of a restricted function with a subclass of its domain. (Contributed by set.mm contributors, 21-Jul-2005.)
(((F A):A–→B C A) → (F C):C–→B)

Theoremfcoi1 5240 Composition of a mapping and restricted identity. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 13-Dec-2003.) (Revised by set.mm contributors, 18-Sep-2011.)
(F:A–→B → (F ( I A)) = F)

Theoremfcoi2 5241 Composition of restricted identity and a mapping. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 13-Dec-2003.) (Revised by set.mm contributors, 18-Sep-2011.)
(F:A–→B → (( I B) F) = F)

Theoremfeu 5242* There is exactly one value of a function in its codomain. (Contributed by set.mm contributors, 10-Dec-2003.)
((F:A–→B C A) → ∃!y B C, y F)

Theoremfcnvres 5243 The converse of a restriction of a function. (Contributed by set.mm contributors, 26-Mar-1998.)
(F:A–→B(F A) = (F B))

Theoremfimacnvdisj 5244 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((F:A–→B (BC) = ) → (FC) = )

Theoremfint 5245* Function into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Oct-1999.) (Revised by set.mm contributors, 18-Sep-2011.)
B       (F:A–→Bx B F:A–→x)

Theoremfin 5246 Mapping into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Sep-1999.) (Revised by set.mm contributors, 18-Sep-2011.)
(F:A–→(BC) ↔ (F:A–→B F:A–→C))

Theoremdmfex 5247 If a mapping is a set, its domain is a set. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 27-Aug-2006.) (Revised by set.mm contributors, 18-Sep-2011.)
((F C F:A–→B) → A V)

Theoremf0 5248 The empty function. (Contributed by set.mm contributors, 14-Aug-1999.)
:–→A

Theoremf00 5249 A class is a function with empty codomain iff it and its domain are empty. (Contributed by set.mm contributors, 10-Dec-2003.)
(F:A–→ ↔ (F = A = ))

Theoremfconst 5250 A cross product with a singleton is a constant function. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Aug-1999.) (Revised by set.mm contributors, 18-Sep-2011.)
B V       (A × {B}):A–→{B}

Theoremfconstg 5251 A cross product with a singleton is a constant function. (Contributed by set.mm contributors, 19-Oct-2004.)
(B V → (A × {B}):A–→{B})

Theoremfnconstg 5252 A cross product with a singleton is a constant function. (Contributed by set.mm contributors, 24-Jul-2014.)
(B V → (A × {B}) Fn A)

Theoremf1eq1 5253 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(F = G → (F:A1-1BG:A1-1B))

Theoremf1eq2 5254 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(A = B → (F:A1-1CF:B1-1C))

Theoremf1eq3 5255 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(A = B → (F:C1-1AF:C1-1B))

Theoremnff1 5256 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
xF    &   xA    &   xB       x F:A1-1B

Theoremdff12 5257* Alternate definition of a one-to-one function. (Contributed by set.mm contributors, 31-Dec-1996.) (Revised by set.mm contributors, 22-Sep-2004.)
(F:A1-1B ↔ (F:A–→B y∃*x xFy))

Theoremf1f 5258 A one-to-one mapping is a mapping. (Contributed by set.mm contributors, 31-Dec-1996.)
(F:A1-1BF:A–→B)

Theoremf1fn 5259 A one-to-one mapping is a function on its domain. (Contributed by set.mm contributors, 8-Mar-2014.)
(F:A1-1BF Fn A)

Theoremf1fun 5260 A one-to-one mapping is a function. (Contributed by set.mm contributors, 8-Mar-2014.)
(F:A1-1B → Fun F)

Theoremf1dm 5261 The domain of a one-to-one mapping. (Contributed by set.mm contributors, 8-Mar-2014.)
(F:A1-1B → dom F = A)

Theoremf1ss 5262 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
((F:A1-1B B C) → F:A1-1C)

Theoremf1funfun 5263 Two ways to express that a set A is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by set.mm contributors, 13-Aug-2004.) (Revised by Scott Fenton, 18-Apr-2021.)
(A:dom A1-1→V ↔ (Fun A Fun A))

Theoremf1co 5264 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 28-May-1998.)
((F:B1-1C G:A1-1B) → (F G):A1-1C)

Theoremfoeq1 5265 Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(F = G → (F:AontoBG:AontoB))

Theoremfoeq2 5266 Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(A = B → (F:AontoCF:BontoC))

Theoremfoeq3 5267 Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(A = B → (F:ContoAF:ContoB))

Theoremnffo 5268 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
xF    &   xA    &   xB       x F:AontoB

Theoremfof 5269 An onto mapping is a mapping. (Contributed by set.mm contributors, 3-Aug-1994.)
(F:AontoBF:A–→B)

Theoremfofun 5270 An onto mapping is a function. (Contributed by set.mm contributors, 29-Mar-2008.)
(F:AontoB → Fun F)

Theoremfofn 5271 An onto mapping is a function on its domain. (Contributed by set.mm contributors, 16-Dec-2008.)
(F:AontoBF Fn A)

Theoremforn 5272 The codomain of an onto function is its range. (Contributed by set.mm contributors, 3-Aug-1994.)
(F:AontoB → ran F = B)

Theoremdffo2 5273 Alternate definition of an onto function. (Contributed by set.mm contributors, 22-Mar-2006.)
(F:AontoB ↔ (F:A–→B ran F = B))

Theoremfoima 5274 The image of the domain of an onto function. (Contributed by set.mm contributors, 29-Nov-2002.)
(F:AontoB → (FA) = B)

Theoremdffn4 5275 A function maps onto its range. (Contributed by set.mm contributors, 10-May-1998.)
(F Fn AF:Aonto→ran F)

Theoremfunforn 5276 A function maps its domain onto its range. (Contributed by set.mm contributors, 23-Jul-2004.)
(Fun AA:dom Aonto→ran A)

Theoremfodmrnu 5277 An onto function has unique domain and range. (Contributed by set.mm contributors, 5-Nov-2006.)
((F:AontoB F:ContoD) → (A = C B = D))

Theoremfores 5278 Restriction of a function. (Contributed by set.mm contributors, 4-Mar-1997.)
((Fun F A dom F) → (F A):Aonto→(FA))

Theoremfoco 5279 Composition of onto functions. (Contributed by set.mm contributors, 22-Mar-2006.)
((F:BontoC G:AontoB) → (F G):AontoC)

Theoremfoconst 5280 A nonzero constant function is onto. (Contributed by set.mm contributors, 12-Jan-2007.)
((F:A–→{B} F) → F:Aonto→{B})

Theoremf1oeq1 5281 Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(F = G → (F:A1-1-ontoBG:A1-1-ontoB))

Theoremf1oeq2 5282 Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(A = B → (F:A1-1-ontoCF:B1-1-ontoC))

Theoremf1oeq3 5283 Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(A = B → (F:C1-1-ontoAF:C1-1-ontoB))

Theoremf1oeq23 5284 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
((A = B C = D) → (F:A1-1-ontoCF:B1-1-ontoD))

Theoremnff1o 5285 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 5286 A one-to-one onto mapping is a one-to-one mapping. (Contributed by set.mm contributors, 12-Dec-2003.)
(F:A1-1-ontoBF:A1-1B)

Theoremf1of 5287 A one-to-one onto mapping is a mapping. (Contributed by set.mm contributors, 12-Dec-2003.)
(F:A1-1-ontoBF:A–→B)

Theoremf1ofn 5288 A one-to-one onto mapping is function on its domain. (Contributed by set.mm contributors, 12-Dec-2003.)
(F:A1-1-ontoBF Fn A)

Theoremf1ofun 5289 A one-to-one onto mapping is a function. (Contributed by set.mm contributors, 12-Dec-2003.)
(F:A1-1-ontoB → Fun F)

Theoremf1odm 5290 The domain of a one-to-one onto mapping. (Contributed by set.mm contributors, 8-Mar-2014.)
(F:A1-1-ontoB → dom F = A)

Theoremdff1o2 5291 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Feb-1997.) (Revised by set.mm contributors, 22-Oct-2011.)
(F:A1-1-ontoB ↔ (F Fn A Fun F ran F = B))

Theoremdff1o3 5292 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 22-Oct-2011.)
(F:A1-1-ontoB ↔ (F:AontoB Fun F))

Theoremf1ofo 5293 A one-to-one onto function is an onto function. (Contributed by set.mm contributors, 28-Apr-2004.)
(F:A1-1-ontoBF:AontoB)

Theoremdff1o4 5294 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 22-Oct-2011.)
(F:A1-1-ontoB ↔ (F Fn A F Fn B))

Theoremdff1o5 5295 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 22-Oct-2011.)
(F:A1-1-ontoB ↔ (F:A1-1B ran F = B))

Theoremf1orn 5296 A one-to-one function maps onto its range. (Contributed by set.mm contributors, 13-Aug-2004.)
(F:A1-1-onto→ran F ↔ (F Fn A Fun F))

Theoremf1f1orn 5297 A one-to-one function maps one-to-one onto its range. (Contributed by set.mm contributors, 4-Sep-2004.)
(F:A1-1BF:A1-1-onto→ran F)

Theoremf1ocnvb 5298 A class is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by set.mm contributors, 8-Dec-2003.) (Modified by Scott Fenton, 17-Apr-2021.)
(F:A1-1-ontoBF:B1-1-ontoA)

Theoremf1ocnv 5299 The converse of a one-to-one onto function is also one-to-one onto. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 11-Feb-1997.) (Revised by set.mm contributors, 22-Oct-2011.)
(F:A1-1-ontoBF:B1-1-ontoA)

Theoremf1ores 5300 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by set.mm contributors, 25-Mar-1998.)
((F:A1-1B C A) → (F C):C1-1-onto→(FC))

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