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Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnniniseg2 5501* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremrexsupp 5502* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremunpreima 5503 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminpreima 5504 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Theoremdifpreima 5505 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)

Theoremrespreima 5506 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiinpreima 5507* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)

Theoremintpreima 5508* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)

Theoremfimacnv 5509 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Theoremsuppss 5510* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremsuppssr 5511 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremfnopfv 5512 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Theoremfvelrn 5513 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Theoremfnfvelrn 5514 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Theoremffvelrn 5515 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Theoremffvelrni 5516 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 5517 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 5518 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 5519* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 5520* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrnmpt 5521* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexrnmpt 5522* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremf0cli 5523 Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)

Theoremdff2 5524 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Theoremdff3 5525* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdff4 5526* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo3 5527* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Theoremdffo4 5528* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo5 5529* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremexfo 5530* A relation equivalent to the existence of an onto mapping. The right-hand is not necessarily a function. (Contributed by NM, 20-Mar-2007.)

Theoremfoelrn 5531* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)

Theoremfoco2 5532 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremfmpt 5533* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremf1ompt 5534* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremfmpti 5535* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremfmptd 5536* Domain and co-domain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremffnfv 5537* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 5538 A function maps to a class to which all values belong. This version of ffnfv 5537 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 5539* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremfmpt2d 5540* Domain and co-domain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremfmpt2dOLD 5541* Domain and co-domain of the mapping operation; deduction form. (Contributed by NM, 9-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremffvresb 5542* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremfmptco 5543* Composition of two functions expressed as ordered-pair class abstractions. If has the equation and the equation then has the equation . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)

Theoremfmptcof 5544* Version of fmptco 5543 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Theoremfmptcos 5545* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfcompt 5546* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremfcoconst 5547 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfsn 5548 A function maps a singleton to a singleton iff it is the singleton of a ordered pair. (Contributed by NM, 10-Dec-2003.)

Theoremfsng 5549 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Theoremfsn2 5550 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

Theoremxpsng 5551 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremxpsn 5552 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)

Theoremdfmpt 5553 Alternate definition for the "maps to" notation df-mpt 3976 (although it requires that be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)

Theoremfnasrn 5554 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremressnop0 5555 If is not in , then the restriction of a singleton of to is null. (Contributed by Scott Fenton, 15-Apr-2011.)

Theoremfpr 5556 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfnressn 5557 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfunressn 5558 A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremfressnfv 5559 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfvconst 5560 The value of a constant function. (Contributed by NM, 30-May-1999.)

Theoremfmptsn 5561* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)

Theoremfmptap 5562* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvresi 5563 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)

Theoremfvunsn 5564 Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremfvsn 5565 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)

Theoremfvsng 5566 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)

Theoremfvsnun1 5567 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5568. (Contributed by NM, 23-Sep-2007.)

Theoremfvsnun2 5568 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5567. (Contributed by NM, 23-Sep-2007.)

Theoremfnsnsplit 5569 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremfsnunf 5570 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunf2 5571 Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunfv 5572 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfsnunres 5573 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremfvpr1 5574 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvpr2 5575 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfvtp1 5576 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp2 5577 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvtp3 5578 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfvconst2g 5579 The value of a constant function. (Contributed by NM, 20-Aug-2005.)

Theoremfconst2g 5580 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)

Theoremfvconst2 5581 The value of a constant function. (Contributed by NM, 16-Apr-2005.)

Theoremfconst2 5582 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)

Theoremfconst5 5583 Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)

Theoremfnsuppres 5584 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Theoremfnsuppeq0 5585 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Theoremfconstfv 5586* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5582. (Contributed by NM, 27-Aug-2004.)

Theoremfconst3 5587 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)

Theoremfconst4 5588 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Theoremresfunexg 5589 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)

TheoremresfunexgALT 5590 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5589 but requires ax-pow 4082. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.)

Theoremcofunexg 5591 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)

Theoremcofunex2g 5592 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)

Theoremfnex 5593 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5589. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

TheoremfnexALT 5594 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5186. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.)

Theoremfunex 5595 If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5593. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)

Theoremopabex 5596* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)

Theoremmptexg 5597* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptex 5598* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremfunrnex 5599 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5595. (Contributed by NM, 11-Nov-1995.)

Theoremzfrep6 5600* A version of the Axiom of Replacement. Normally would have free variables and . Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4038 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 4028. (Contributed by NM, 10-Oct-2003.)

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