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Theorem List for New Foundations Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremf1orescnv 5301 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremf1imacnv 5302 Preimage of an image. (Contributed by set.mm contributors, 30-Sep-2004.)

Theoremfoimacnv 5303 A reverse version of f1imacnv 5302. (Contributed by Jeffrey Hankins, 16-Jul-2009.)

Theoremf1oun 5304 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by set.mm contributors, 26-Mar-1998.)

Theoremfun11iun 5305* 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.)

Theoremresdif 5306 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.)

Theoremresin 5307 The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremf1oco 5308 Composition of one-to-one onto functions. (Contributed by set.mm contributors, 19-Mar-1998.)

Theoremf1ococnv2 5309 The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by set.mm contributors, 13-Dec-2003.)

Theoremf1ococnv1 5310 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 set.mm contributors, 13-Dec-2003.)

Theoremf1cnv 5311 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)

Theoremf1cocnv1 5312 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremf1cocnv2 5313 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremffoss 5314* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by set.mm contributors, 10-May-1998.)

Theoremf11o 5315* Relationship between one-to-one and one-to-one onto function. (Contributed by set.mm contributors, 4-Apr-1998.)

Theoremf10 5316 The empty set maps one-to-one into any class. (Contributed by set.mm contributors, 7-Apr-1998.)

Theoremf1o00 5317 One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 15-Apr-1998.)

Theoremfo00 5318 Onto mapping of the empty set. (Contributed by set.mm contributors, 22-Mar-2006.)

Theoremf1o0 5319 One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 10-Feb-2004.) (Revised by set.mm contributors, 16-Feb-2004.)

Theoremf1oi 5320 A restriction of the identity relation is a one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 30-Apr-1998.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremf1ovi 5321 The identity relation is a one-to-one onto function on the universe. (Contributed by set.mm contributors, 16-May-2004.)

Theoremf1osn 5322 A singleton of an ordered pair is one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 18-May-1998.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremf1osng 5323 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremfv2 5324* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 30-Apr-2004.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfvprc 5325 A function's value at a proper class is the empty set. (Contributed by set.mm contributors, 20-May-1998.)

Theoremelfv 5326* Membership in a function value. (Contributed by set.mm contributors, 30-Apr-2004.)

Theoremfveq1 5327 Equality theorem for function value. (Contributed by set.mm contributors, 29-Dec-1996.)

Theoremfveq2 5328 Equality theorem for function value. (Contributed by set.mm contributors, 29-Dec-1996.)

Theoremfveq1i 5329 Equality inference for function value. (Contributed by set.mm contributors, 2-Sep-2003.)

Theoremfveq1d 5330 Equality deduction for function value. (Contributed by set.mm contributors, 2-Sep-2003.)

Theoremfveq2i 5331 Equality inference for function value. (Contributed by set.mm contributors, 28-Jul-1999.)

Theoremfveq2d 5332 Equality deduction for function value. (Contributed by set.mm contributors, 29-May-1999.)

Theoremfveq12d 5333 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)

Theoremnffv 5334 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnffvd 5335 Deduction version of bound-variable hypothesis builder nffv 5334. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbfv12g 5336 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)

Theoremcsbfv2g 5337* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)

Theoremcsbfvg 5338* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)

Theoremfvex 5339 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Dec-1996.)

Theoremfvif 5340 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfv3 5341* 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.)

Theoremfvres 5342 The value of a restricted function. (Contributed by set.mm contributors, 2-Aug-1994.) (Revised by set.mm contributors, 16-Feb-2004.)

Theoremfunssfv 5343 The value of a member of the domain of a subclass of a function. (Contributed by set.mm contributors, 15-Aug-1994.) (Revised by set.mm contributors, 29-May-2007.)

Theoremtz6.12-1 5344* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12 5345* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)

Theoremtz6.12-2 5346* Function value when is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Apr-2004.)

Theoremtz6.12c 5347* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12i 5348 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Apr-2004.) (Revised by set.mm contributors, 6-Apr-2007.)

Theoremndmfv 5349 The value of a class outside its domain is the empty set. (Contributed by set.mm contributors, 24-Aug-1995.)

Theoremndmfvrcl 5350 Reverse closure law for function with the empty set not in its domain. (Contributed by set.mm contributors, 26-Apr-1996.)

Theoremelfvdm 5351 If a function value has a member, the argument belongs to the domain. (Contributed by set.mm contributors, 12-Feb-2007.)

Theoremnfvres 5352 The value of a non-member of a restriction is the empty set. (Contributed by set.mm contributors, 13-Nov-1995.)

Theoremnfunsn 5353 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfv01 5354 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Theoremfveqres 5355 Equal values imply equal values in a restriction. (Contributed by set.mm contributors, 13-Nov-1995.)

Theoremfunbrfv 5356 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfunopfv 5357 The second element in an ordered pair member of a function is the function's value. (Contributed by set.mm contributors, 19-Jul-1996.)

Theoremfnbrfvb 5358 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfnopfvb 5359 Equivalence of function value and ordered pair membership. (Contributed by set.mm contributors, 9-Jan-2015.)

Theoremfunbrfvb 5360 Equivalence of function value and binary relation. (Contributed by set.mm contributors, 9-Jan-2015.)

Theoremfunopfvb 5361 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by set.mm contributors, 9-Jan-2015.)

Theoremfunbrfv2b 5362 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremdffn5 5363* Representation of a function in terms of its values. (Contributed by set.mm contributors, 29-Jan-2004.)

Theoremfnrnfv 5364* The range of a function expressed as a collection of the function's values. (Contributed by set.mm contributors, 20-Oct-2005.)

Theoremfvelrnb 5365* A member of a function's range is a value of the function. (Contributed by set.mm contributors, 31-Oct-1995.)

Theoremdfimafn 5366* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremdfimafn2 5367* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass4 5368* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfvelima 5369* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 29-Apr-2004.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremfvelimab 5370* Function value in an image. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (An unnecessary distinct variable restriction was removed by David Abernethy, 17-Dec-2011.) (Contributed by set.mm contributors, 20-Jan-2007.) (Revised by set.mm contributors, 25-Dec-2011.)

Theoremfniniseg 5371 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremfniinfv 5372* The indexed intersection of a function's values is the intersection of its range. (Contributed by set.mm contributors, 20-Oct-2005.)

Theoremfnsnfv 5373 Singleton of function value. (Contributed by set.mm contributors, 22-May-1998.)

Theoremfnimapr 5374 The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremfunfv 5375 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)

Theoremfunfv2 5376* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by set.mm contributors, 22-May-1998.) (Revised by set.mm contributors, 11-May-2005.)

Theoremfunfv2f 5377 The value of a function. Version of funfv2 5376 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)

Theoremfvun 5378 Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)

Theoremfvun1 5379 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremfvun2 5380 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremdmfco 5381 Domains of a function composition. (Contributed by set.mm contributors, 27-Jan-1997.)

Theoremfvco2 5382 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 9-Oct-2004.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremfvco 5383 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by set.mm contributors, 22-Apr-2006.)

Theoremfvco3 5384 Value of a function composition. (Contributed by set.mm contributors, 3-Jan-2004.) (Revised by set.mm contributors, 21-Aug-2006.)

Theoremfvopab4t 5385* Closed theorem form of fvopab4 5389. (Contributed by set.mm contributors, 21-Feb-2013.)

Theoremfvopab3g 5386* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 6-Mar-1996.)

Theoremfvopab3ig 5387* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)

Theoremfvopab4g 5388* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)

Theoremfvopab4 5389* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)

Theoremfvopab4ndm 5390* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by set.mm contributors, 28-Mar-2008.)

Theoremfvopabg 5391* The value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 2-Sep-2003.)

Theoremeqfnfv 5392* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremeqfnfv2 5393* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 5-Feb-2004.)

Theoremeqfnfv3 5394* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5395* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5396* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5392 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5397* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5398* Equality of restricted functions is determined by their values. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 6-Feb-2004.)

Theoremchfnrn 5399* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by set.mm contributors, 31-Aug-1999.)

Theoremfunfvop 5400 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by set.mm contributors, 14-Oct-1996.)

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