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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvtp1g 5701 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp2g 5702 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp3g 5703 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp1 5704 | The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvtp2 5705 | The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvtp3 5706 | The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvconst2g 5707 | The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
Theorem | fconst2g 5708 | A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.) |
Theorem | fvconst2 5709 | The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
Theorem | fconst2 5710 | A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.) |
Theorem | fconstfvm 5711* | A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5710. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Theorem | fconst3m 5712* | Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Theorem | fconst4m 5713* | Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
Theorem | resfunexg 5714 | 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.) |
Theorem | fnex 5715 | 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 5714. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | funex 5716 | If the domain of a function exists, so does 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 5715. (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.) |
Theorem | opabex 5717* | Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
Theorem | mptexg 5718* | 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.) |
Theorem | mptex 5719* | 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.) |
Theorem | mptexd 5720* | If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 5718. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Theorem | mptrabex 5721* | If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.) |
Theorem | fex 5722 | If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.) |
Theorem | eufnfv 5723* | A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
Theorem | funfvima 5724 | A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
Theorem | funfvima2 5725 | A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
Theorem | funfvima3 5726 | A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.) |
Theorem | fnfvima 5727 | The function value of an operand in a set is contained in the image of that set, using the abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.) |
Theorem | foima2 5728* | Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5423). (Contributed by BJ, 6-Jul-2022.) |
Theorem | foelrn 5729* | Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.) |
Theorem | foco2 5730 | If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Theorem | rexima 5731* | Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | ralima 5732* | Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | idref 5733* |
TODO: This is the same as issref 4991 (which has a much longer proof).
Should we replace issref 4991 with this one? - NM 9-May-2016.
Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.) |
Theorem | elabrex 5734* | Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Theorem | abrexco 5735* | Composition of two image maps and . (Contributed by NM, 27-May-2013.) |
Theorem | imaiun 5736* | The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Theorem | imauni 5737* | The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) |
Theorem | fniunfv 5738* | The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.) |
Theorem | funiunfvdm 5739* | The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5738. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | funiunfvdmf 5740* | The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5739 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | eluniimadm 5741* | Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | elunirn 5742* | Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.) |
Theorem | fnunirn 5743* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Theorem | dff13 5744* | A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.) |
Theorem | f1veqaeq 5745 | If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
Theorem | dff13f 5746* | A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.) |
Theorem | f1mpt 5747* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | f1fveq 5748 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
Theorem | f1elima 5749 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | f1imass 5750 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | f1imaeq 5751 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | dff1o6 5752* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
Theorem | f1ocnvfv1 5753 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv2 5754 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv 5755 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | f1ocnvfvb 5756 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvdm 5757 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfvrneq 5758 | If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
Theorem | fcof1 5759 | An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Theorem | fcofo 5760 | An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
Theorem | cbvfo 5761* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Theorem | cbvexfo 5762* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
Theorem | cocan1 5763 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Theorem | cocan2 5764 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Theorem | fcof1o 5765 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Theorem | foeqcnvco 5766 | Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Theorem | f1eqcocnv 5767 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Theorem | fliftrel 5768* | , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel 5769* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel1 5770* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftcnv 5771* | Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfun 5772* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfund 5773* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfuns 5774* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftf 5775* | The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftval 5776* | The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | isoeq1 5777 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq2 5778 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq3 5779 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq4 5780 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq5 5781 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | nfiso 5782 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | isof1o 5783 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
Theorem | isorel 5784 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
Theorem | isoresbr 5785* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Theorem | isoid 5786 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv 5787 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv2 5788 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
Theorem | isores2 5789 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores1 5790 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores3 5791 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
Theorem | isotr 5792 | Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Theorem | iso0 5793 | The empty set is an isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
Theorem | isoini 5794 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
Theorem | isoini2 5795 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
Theorem | isoselem 5796* | Lemma for isose 5797. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isose 5797 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isopolem 5798 | Lemma for isopo 5799. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isopo 5799 | An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isosolem 5800 | Lemma for isoso 5801. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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