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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvconst 5601 | The value of a constant function. (Contributed by NM, 30-May-1999.) |
Theorem | fmptsn 5602* | 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.) |
Theorem | fmptap 5603* | Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | fmptapd 5604* | Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Theorem | fmptpr 5605* | Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Theorem | fvresi 5606 | The value of a restricted identity function. (Contributed by NM, 19-May-2004.) |
Theorem | fvunsng 5607 | Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.) |
Theorem | fvsn 5608 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.) |
Theorem | fvsng 5609 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) |
Theorem | fvsnun1 5610 | The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5611. (Contributed by NM, 23-Sep-2007.) |
Theorem | fvsnun2 5611 | The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5610. (Contributed by NM, 23-Sep-2007.) |
Theorem | fnsnsplitss 5612 | Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.) |
Theorem | fsnunf 5613 | Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | fsnunfv 5614 | Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.) |
Theorem | fsnunres 5615 | Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | funresdfunsnss 5616 | Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.) |
Theorem | fvpr1 5617 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Theorem | fvpr2 5618 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Theorem | fvpr1g 5619 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Theorem | fvpr2g 5620 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Theorem | fvtp1g 5621 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp2g 5622 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp3g 5623 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp1 5624 | The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvtp2 5625 | The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvtp3 5626 | The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvconst2g 5627 | The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
Theorem | fconst2g 5628 | A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.) |
Theorem | fvconst2 5629 | The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
Theorem | fconst2 5630 | A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.) |
Theorem | fconstfvm 5631* | A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5630. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Theorem | fconst3m 5632* | Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Theorem | fconst4m 5633* | Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
Theorem | resfunexg 5634 | 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 5635 | 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 5634. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | funex 5636 | 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 5635. (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 5637* | Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
Theorem | mptexg 5638* | 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 5639* | 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 | fex 5640 | If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.) |
Theorem | eufnfv 5641* | A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
Theorem | funfvima 5642 | A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
Theorem | funfvima2 5643 | A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
Theorem | funfvima3 5644 | 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 5645 | 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 5646* | 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 5345). (Contributed by BJ, 6-Jul-2022.) |
Theorem | foelrn 5647* | Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.) |
Theorem | foco2 5648 | 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 5649* | Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | ralima 5650* | Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | idref 5651* |
TODO: This is the same as issref 4916 (which has a much longer proof).
Should we replace issref 4916 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 5652* | Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Theorem | abrexco 5653* | Composition of two image maps and . (Contributed by NM, 27-May-2013.) |
Theorem | imaiun 5654* | The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Theorem | imauni 5655* | 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 5656* | 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 5657* | 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 5656. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | funiunfvdmf 5658* | The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5657 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | eluniimadm 5659* | Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | elunirn 5660* | Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.) |
Theorem | fnunirn 5661* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Theorem | dff13 5662* | 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 5663 | 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 5664* | 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 5665* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | f1fveq 5666 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
Theorem | f1elima 5667 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | f1imass 5668 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | f1imaeq 5669 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | dff1o6 5670* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
Theorem | f1ocnvfv1 5671 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv2 5672 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv 5673 | 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 5674 | 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 5675 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfvrneq 5676 | 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 5677 | 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 5678 | 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 5679* | 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 5680* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
Theorem | cocan1 5681 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Theorem | cocan2 5682 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Theorem | fcof1o 5683 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Theorem | foeqcnvco 5684 | 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 5685 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Theorem | fliftrel 5686* | , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel 5687* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel1 5688* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftcnv 5689* | Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfun 5690* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfund 5691* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfuns 5692* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftf 5693* | The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftval 5694* | The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | isoeq1 5695 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq2 5696 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq3 5697 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq4 5698 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq5 5699 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | nfiso 5700 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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