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

Theoremfuncnvres 5201 The converse of a restricted function. (Contributed by set.mm contributors, 27-Mar-1998.)
(Fun F(F A) = (F (FA)))

Theoremcnvresid 5202 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
( I A) = ( I A)

Theoremfuncnvres2 5203 The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by set.mm contributors, 4-May-2005.)
(Fun F(F A) = (F (FA)))

Theoremfunimacnv 5204 The image of the preimage of a function. (Contributed by set.mm contributors, 25-May-2004.)
(Fun F → (F “ (FA)) = (A ∩ ran F))

Theoremfunimass1 5205 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by set.mm contributors, 25-May-2004.)
((Fun F A ran F) → ((FA) BA (FB)))

Theoremfunimass2 5206 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by set.mm contributors, 25-May-2004.) (Revised by set.mm contributors, 4-May-2007.)
((Fun F A (FB)) → (FA) B)

Theoremimadif 5207 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
(Fun F → (F “ (A B)) = ((FA) (FB)))

Theoremimain 5208 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
(Fun F → (F “ (AB)) = ((FA) ∩ (FB)))

Theoremfneq1 5209 Equality theorem for function predicate with domain. (Contributed by set.mm contributors, 1-Aug-1994.)
(F = G → (F Fn AG Fn A))

Theoremfneq2 5210 Equality theorem for function predicate with domain. (Contributed by set.mm contributors, 1-Aug-1994.)
(A = B → (F Fn AF Fn B))

Theoremfneq1d 5211 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(φF = G)       (φ → (F Fn AG Fn A))

Theoremfneq2d 5212 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (F Fn AF Fn B))

Theoremfneq12d 5213 Equality deduction for function predicate with domain. (Contributed by set.mm contributors, 26-Jun-2011.)
(φF = G)    &   (φA = B)       (φ → (F Fn AG Fn B))

Theoremfneq1i 5214 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
F = G       (F Fn AG Fn A)

Theoremfneq2i 5215 Equality inference for function predicate with domain. (Contributed by set.mm contributors, 4-Sep-2011.)
A = B       (F Fn AF Fn B)

Theoremnffn 5216 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
F/_xF    &   F/_xA       x F Fn A

Theoremfnfun 5217 A function with domain is a function. (Contributed by set.mm contributors, 1-Aug-1994.)
(F Fn A → Fun F)

Theoremfnrel 5218 A function with domain is a relation. (Contributed by set.mm contributors, 1-Aug-1994.)
(F Fn A → Rel F)

Theoremfndm 5219 The domain of a function. (Contributed by set.mm contributors, 2-Aug-1994.)
(F Fn A → dom F = A)

Theoremfunfni 5220 Inference to convert a function and domain antecedent. (Contributed by set.mm contributors, 22-Apr-2004.)
((Fun F B dom F) → φ)       ((F Fn A B A) → φ)

Theoremfndmu 5221 A function has a unique domain. (Contributed by set.mm contributors, 11-Aug-1994.)
((F Fn A F Fn B) → A = B)

Theoremfnbr 5222 The first argument of binary relation on a function belongs to the function's domain. (Contributed by set.mm contributors, 7-May-2004.)
((F Fn A BFC) → B A)

Theoremfnop 5223 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by set.mm contributors, 8-Aug-1994.) (Revised by set.mm contributors, 25-Mar-2007.)
((F Fn A B, C F) → B A)

Theoremfneu 5224* There is exactly one value of a function. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 22-Apr-2004.) (Revised by set.mm contributors, 18-Sep-2011.)
((F Fn A B A) → ∃!y BFy)

Theoremfneu2 5225* There is exactly one value of a function. (Contributed by set.mm contributors, 7-Nov-1995.)
((F Fn A B A) → ∃!yB, y F)

Theoremfnun 5226 The union of two functions with disjoint domains. (Contributed by set.mm contributors, 22-Sep-2004.)
(((F Fn A G Fn B) (AB) = ) → (FG) Fn (AB))

Theoremfnunsn 5227 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.)
(φX V)    &   (φY V)    &   (φF Fn D)    &   G = (F ∪ {X, Y})    &   E = (D ∪ {X})    &   (φ → ¬ X D)       (φG Fn E)

Theoremfnco 5228 Composition of two functions. (Contributed by set.mm contributors, 22-May-2006.)
((F Fn A G Fn B ran G A) → (F G) Fn B)

Theoremfnresdm 5229 A function does not change when restricted to its domain. (Contributed by set.mm contributors, 5-Sep-2004.)
(F Fn A → (F A) = F)

Theoremfnresdisj 5230 A function restricted to a class disjoint with its domain is empty. (Contributed by set.mm contributors, 23-Sep-2004.)
(F Fn A → ((AB) = ↔ (F B) = ))

Theorem2elresin 5231 Membership in two functions restricted by each other's domain. (Contributed by set.mm contributors, 8-Aug-1994.)
((F Fn A G Fn B) → ((x, y F x, z G) ↔ (x, y (F (AB)) x, z (G (AB)))))

Theoremfnssresb 5232 Restriction of a function with a subclass of its domain. (Contributed by set.mm contributors, 10-Oct-2007.)
(F Fn A → ((F B) Fn BB A))

Theoremfnssres 5233 Restriction of a function with a subclass of its domain. (Contributed by set.mm contributors, 2-Aug-1994.) (Revised by set.mm contributors, 25-Sep-2004.)
((F Fn A B A) → (F B) Fn B)

Theoremfnresin1 5234 Restriction of a function's domain with an intersection. (Contributed by set.mm contributors, 9-Aug-1994.)
(F Fn A → (F (AB)) Fn (AB))

Theoremfnresin2 5235 Restriction of a function's domain with an intersection. (Contributed by set.mm contributors, 9-Aug-1994.)
(F Fn A → (F (BA)) Fn (BA))

Theoremfnres 5236* An equivalence for functionality of a restriction. Compare dffun8 5168. (Contributed by Mario Carneiro, 20-May-2015.)
((F A) Fn Ax A ∃!y xFy)

Theoremfnresi 5237 Functionality and domain of restricted identity. (Contributed by set.mm contributors, 27-Aug-2004.)
( I A) Fn A

Theoremfnima 5238 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 5239 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 5240 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 5241* 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 5242* 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 5243* 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 5244* 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 5245* 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 5246* 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 5247 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(F = G → (F:A–→BG:A–→B))

Theoremfeq2 5248 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(A = B → (F:A–→CF:B–→C))

Theoremfeq3 5249 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)
(A = B → (F:C–→AF:C–→B))

Theoremfeq23 5250 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 5251 Equality deduction for functions. (Contributed by set.mm contributors, 19-Feb-2008.)
(φF = G)       (φ → (F:A–→BG:A–→B))

Theoremfeq2d 5252 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (F:A–→CF:B–→C))

Theoremfeq12d 5253 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φF = G)    &   (φA = B)       (φ → (F:A–→CG:B–→C))

Theoremfeq1i 5254 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
F = G       (F:A–→BG:A–→B)

Theoremfeq2i 5255 Equality inference for functions. (Contributed by set.mm contributors, 5-Sep-2011.)
A = B       (F:A–→CF:B–→C)

Theoremfeq23i 5256 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
A = C    &   B = D       (F:A–→BF:C–→D)

Theoremfeq23d 5257 Equality deduction for functions. (Contributed by set.mm contributors, 8-Jun-2013.)
(φA = C)    &   (φB = D)       (φ → (F:A–→BF:C–→D))

Theoremnff 5258 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
F/_xF    &   F/_xA    &   F/_xB       x F:A–→B

Theoremelimf 5259 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 5260 A mapping is a function. (Contributed by set.mm contributors, 2-Aug-1994.)
(F:A–→BF Fn A)

Theoremdffn2 5261 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 5262 A mapping is a function. (Contributed by set.mm contributors, 3-Aug-1994.)
(F:A–→B → Fun F)

Theoremfrel 5263 A mapping is a relation. (Contributed by set.mm contributors, 3-Aug-1994.)
(F:A–→B → Rel F)

Theoremfdm 5264 The domain of a mapping. (Contributed by set.mm contributors, 2-Aug-1994.)
(F:A–→B → dom F = A)

Theoremfdmi 5265 The domain of a mapping. (Contributed by set.mm contributors, 28-Jul-2008.)
F:A–→B       dom F = A

Theoremfrn 5266 The range of a mapping. (Contributed by set.mm contributors, 3-Aug-1994.)
(F:A–→B → ran F B)

Theoremdffn3 5267 A function maps to its range. (Contributed by set.mm contributors, 1-Sep-1999.)
(F Fn AF:A–→ran F)

Theoremfss 5268 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 5269 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 5270 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 5271 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 5272 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 5273 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 5274 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 5275 Composition of two functions. (Contributed by set.mm contributors, 22-May-2006.)
((F Fn A G:B–→A) → (F G) Fn B)

Theoremfssres 5276 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 5277 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 5278 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 5279 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 5280* 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 5281 The converse of a restriction of a function. (Contributed by set.mm contributors, 26-Mar-1998.)
(F:A–→B(F A) = (F B))

Theoremfimacnvdisj 5282 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 5283* 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 5284 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 5285 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 5286 The empty function. (Contributed by set.mm contributors, 14-Aug-1999.)
:–→A

Theoremf00 5287 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 5288 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 5289 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 5290 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 5291 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(F = G → (F:A1-1BG:A1-1B))

Theoremf1eq2 5292 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(A = B → (F:A1-1CF:B1-1C))

Theoremf1eq3 5293 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)
(A = B → (F:C1-1AF:C1-1B))

Theoremnff1 5294 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
F/_xF    &   F/_xA    &   F/_xB       x F:A1-1B

Theoremdff12 5295* 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 5296 A one-to-one mapping is a mapping. (Contributed by set.mm contributors, 31-Dec-1996.)
(F:A1-1BF:A–→B)

Theoremf1fn 5297 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 5298 A one-to-one mapping is a function. (Contributed by set.mm contributors, 8-Mar-2014.)
(F:A1-1B → Fun F)

Theoremf1rel 5299 A one-to-one onto mapping is a relation. (Contributed by set.mm contributors, 8-Mar-2014.)
(F:A1-1B → Rel F)

Theoremf1dm 5300 The domain of a one-to-one mapping. (Contributed by set.mm contributors, 8-Mar-2014.)
(F:A1-1B → dom F = A)

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