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Theorem List for Metamath Proof Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremf1rel 5601 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremf1dm 5602 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)

Theoremf1ss 5603 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.)

Theoremf1ssr 5604 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremf1ssres 5605 A function that is one-to-one is also one-to-one on some aubset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremf1cnvcnv 5606 Two ways to express that a set (not necessarily a function) 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 NM, 13-Aug-2004.)

Theoremf1co 5607 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)

Theoremfoeq1 5608 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq2 5609 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq3 5610 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremnffo 5611 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Theoremfof 5612 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremfofun 5613 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)

Theoremfofn 5614 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)

Theoremforn 5615 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)

Theoremdffo2 5616 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)

Theoremfoima 5617 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Theoremdffn4 5618 A function maps onto its range. (Contributed by NM, 10-May-1998.)

Theoremfunforn 5619 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)

Theoremfodmrnu 5620 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)

Theoremfores 5621 Restriction of a function. (Contributed by NM, 4-Mar-1997.)

Theoremfoco 5622 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

Theoremfoconst 5623 A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)

Theoremf1oeq1 5624 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq2 5625 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq3 5626 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq23 5627 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Theoremf1eq123d 5628 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremfoeq123d 5629 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremf1oeq123d 5630 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremnff1o 5631 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)

Theoremf1of1 5632 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1of 5633 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofn 5634 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofun 5635 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)

Theoremf1orel 5636 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)

Theoremf1odm 5637 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)

Theoremdff1o2 5638 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o3 5639 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ofo 5640 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)

Theoremdff1o4 5641 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o5 5642 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1orn 5643 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Theoremf1f1orn 5644 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)

Theoremf1oabexg 5645* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremf1ocnv 5646 The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ocnvb 5647 A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)

Theoremf1ores 5648 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)

Theoremf1orescnv 5649 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremf1imacnv 5650 Preimage of an image. (Contributed by NM, 30-Sep-2004.)

Theoremfoimacnv 5651 A reverse version of f1imacnv 5650. (Contributed by Jeffrey Hankins, 16-Jul-2009.)

Theoremfoun 5652 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremf1oun 5653 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Theoremfun11iun 5654* 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 5655 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 5656 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 5657 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)

Theoremf1cnv 5658 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)

Theoremfuncocnv2 5659 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfococnv2 5660 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1ococnv2 5661 The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)

Theoremf1cocnv2 5662 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremf1ococnv1 5663 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 NM, 13-Dec-2003.)

Theoremf1cocnv1 5664 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremfuncoeqres 5665 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremffoss 5666* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)

Theoremf11o 5667* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)

Theoremf10 5668 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)

Theoremf1o00 5669 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)

Theoremfo00 5670 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremf1o0 5671 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)

Theoremf1oi 5672 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ovi 5673 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)

Theoremf1osn 5674 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1osng 5675 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1oprswap 5676 A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremf1oprg 5677 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5676. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theoremtz6.12-2 5678* Function value when is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfveu 5679* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)

Theorembrprcneu 5680* If is a proper class, then there is no unique binary relationship with as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)

Theoremfvprc 5681 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)

Theoremfv2 5682* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdffv3 5683* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdffv4 5684* The previous definition of function value, from before the operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5191), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Theoremelfv 5685* Membership in a function value. (Contributed by NM, 30-Apr-2004.)

Theoremfveq1 5686 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq2 5687 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq1i 5688 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq1d 5689 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq2i 5690 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)

Theoremfveq2d 5691 Equality deduction for function value. (Contributed by NM, 29-May-1999.)

Theoremfveq12i 5692 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)

Theoremfveq12d 5693 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)

Theoremnffv 5694 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnffvmpt1 5695* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremnffvd 5696 Deduction version of bound-variable hypothesis builder nffv 5694. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbfv12g 5697 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)

Theoremcsbfv12gALT 5698 Move class substitution in and out of a function value.(This is csbfv12g 5697 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbfv12gALTVD 28720. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcsbfv2g 5699* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)

Theoremcsbfvg 5700* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)

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