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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremf1oeq2 5701 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq3 5702 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq23 5703 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Theoremf1eq123d 5704 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremfoeq123d 5705 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremf1oeq123d 5706 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremnff1o 5707 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)

Theoremf1of1 5708 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1of 5709 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofn 5710 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofun 5711 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)

Theoremf1orel 5712 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)

Theoremf1odm 5713 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)

Theoremdff1o2 5714 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o3 5715 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ofo 5716 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)

Theoremdff1o4 5717 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o5 5718 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1orn 5719 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Theoremf1f1orn 5720 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)

Theoremf1oabexg 5721* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremf1ocnv 5722 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 5723 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 5724 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)

Theoremf1orescnv 5725 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremf1imacnv 5726 Preimage of an image. (Contributed by NM, 30-Sep-2004.)

Theoremfoimacnv 5727 A reverse version of f1imacnv 5726. (Contributed by Jeffrey Hankins, 16-Jul-2009.)

Theoremfoun 5728 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremf1oun 5729 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Theoremfun11iun 5730* 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 5731 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 5732 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 5733 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)

Theoremf1cnv 5734 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)

Theoremfuncocnv2 5735 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfococnv2 5736 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1ococnv2 5737 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 5738 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremf1ococnv1 5739 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 5740 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremfuncoeqres 5741 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremffoss 5742* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)

Theoremf11o 5743* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)

Theoremf10 5744 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)

Theoremf1o00 5745 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)

Theoremfo00 5746 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremf1o0 5747 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)

Theoremf1oi 5748 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 5749 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)

Theoremf1osn 5750 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 5751 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1oprswap 5752 A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremf1oprg 5753 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5752. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theoremtz6.12-2 5754* 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 5755* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)

Theorembrprcneu 5756* 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 5757 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)

Theoremfv2 5758* 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 5759* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdffv4 5760* 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 5267), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Theoremelfv 5761* Membership in a function value. (Contributed by NM, 30-Apr-2004.)

Theoremfveq1 5762 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq2 5763 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq1i 5764 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq1d 5765 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq2i 5766 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)

Theoremfveq2d 5767 Equality deduction for function value. (Contributed by NM, 29-May-1999.)

Theoremfveq12i 5768 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)

Theoremfveq12d 5769 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)

Theoremnffv 5770 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnffvmpt1 5771* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremnffvd 5772 Deduction version of bound-variable hypothesis builder nffv 5770. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvex 5773 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)

Theoremfvif 5774 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfv3 5775* 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 5776 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)

Theoremfunssfv 5777 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)

Theoremtz6.12-1 5778* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12 5779* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)

Theoremtz6.12f 5780* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)

Theoremtz6.12c 5781* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12i 5782 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremfvbr0 5783 Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfvrn0 5784 A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)

Theoremfvssunirn 5785 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremndmfv 5786 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmfvrcl 5787 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)

Theoremelfvdm 5788 If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.)

Theoremelfvex 5789 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremelfvexd 5790 If a function value is nonempty, its argument is a set. Deduction form of elfvex 5789. (Contributed by David Moews, 1-May-2017.)

Theoremnfvres 5791 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Theoremnfunsn 5792 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.)

Theoremfvfundmfvn0 5793 If a class's value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)

Theoremfv01 5794 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Theoremfveqres 5795 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)

Theoremcsbfv12 5796 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)

Theoremcsbfv12gOLD 5797 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) Obsolete as of 20-Aug-2018. Use csbfv12 5796 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremcsbfv2g 5798* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)

Theoremcsbfv 5799* Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)

TheoremcsbfvgOLD 5800* Substitution for a function value. (Contributed by NM, 1-Jan-2006.) Obsolete as of 20-Aug-2018. Use csbfv 5799 instead. (New usage is discouraged.) (Proof modification is discouraged.)

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