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Type | Label | Description |
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Statement | ||
Theorem | fsnunres 5901 | Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
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Theorem | fvpr1 5902 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
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Theorem | fvpr2 5903 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
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Theorem | fvpr1g 5904 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
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Theorem | fvpr2g 5905 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
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Theorem | fvtp1 5906 | The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
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Theorem | fvtp2 5907 | The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
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Theorem | fvtp3 5908 | The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
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Theorem | fvtp1g 5909 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
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Theorem | fvtp2g 5910 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
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Theorem | fvtp3g 5911 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
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Theorem | fvconst2g 5912 | The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
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Theorem | fconst2g 5913 | A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.) |
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Theorem | fvconst2 5914 | The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
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Theorem | fconst2 5915 | A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.) |
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Theorem | fconst5 5916 | Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.) |
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Theorem | fnpr 5917 | Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) (Revised by NM, 10-Dec-2017.) |
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Theorem | fnprOLD 5918 | Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Obsolete version of fnpr 5917 as of 10-Dec-2017. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | fnsuppres 5919 | Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) |
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Theorem | fnsuppeq0 5920 | The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
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Theorem | fconstfv 5921* | A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5915. (Contributed by NM, 27-Aug-2004.) |
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Theorem | fconst3 5922 | Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.) |
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Theorem | fconst4 5923 | Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
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Theorem | resfunexg 5924 | 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.) |
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Theorem | resfunexgALT 5925 | 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 5924 but requires ax-pow 4345. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | cofunexg 5926 | Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.) |
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Theorem | cofunex2g 5927 | Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.) |
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Theorem | fnex 5928 | 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 5924. See fnexALT 5929 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
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Theorem | fnexALT 5929 | 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 5497. This version of fnex 5928 uses ax-pow 4345, whereas fnex 5928 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | funex 5930 | 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 5928. (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.) |
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Theorem | opabex 5931* | Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
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Theorem | mptexg 5932* | 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.) |
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Theorem | mptex 5933* | 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.) |
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Theorem | funrnex 5934 | 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 5930. (Contributed by NM, 11-Nov-1995.) |
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Theorem | zfrep6 5935* |
A version of the Axiom of Replacement. Normally ![]() ![]() ![]() |
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Theorem | fex 5936 | If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.) |
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Theorem | fornex 5937 | If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.) |
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Theorem | f1dmex 5938 | If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4288. (Contributed by NM, 4-Sep-2004.) |
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Theorem | eufnfv 5939* | A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
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Theorem | funfvima 5940 | A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
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Theorem | funfvima2 5941 | A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
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Theorem | funfvima3 5942 | 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.) |
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Theorem | fnfvima 5943 |
The function value of an operand in a set is contained in the image of
that set, using the ![]() |
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Theorem | rexima 5944* | Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
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Theorem | ralima 5945* | Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
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Theorem | idref 5946* |
TODO: This is the same as issref 5214 (which has a much longer proof).
Should we replace issref 5214 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.) |
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Theorem | fvclss 5947* | Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.) |
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Theorem | fvclex 5948* | Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.) |
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Theorem | fvresex 5949* | Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.) |
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Theorem | abrexex 5950* |
Existence of a class abstraction of existentially restricted sets. ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | abrexexg 5951* |
Existence of a class abstraction of existentially restricted sets. ![]() ![]() ![]() |
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Theorem | elabrex 5952* | Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
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Theorem | abrexco 5953* |
Composition of two image maps ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | iunexg 5954* |
The existence of an indexed union. ![]() ![]() |
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Theorem | abrexex2g 5955* | Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | opabex3d 5956* | Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) |
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Theorem | opabex3 5957* | Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | iunex 5958* |
The existence of an indexed union. ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | imaiun 5959* | The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
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Theorem | imauni 5960* | 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.) |
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Theorem | fniunfv 5961* | 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.) |
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Theorem | funiunfv 5962* |
The indexed union of a function's values is the union of its image under
the index class.
Note: This theorem depends on the fact that our function value is the
empty set outside of its domain. If the antecedent is changed to
|
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Theorem | funiunfvf 5963* | The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5962 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.) |
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Theorem | eluniima 5964* | Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.) |
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Theorem | elunirn 5965* | Membership in the union of the range of a function. See elunirnALT 5967 for alternate proof. (Contributed by NM, 24-Sep-2006.) |
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Theorem | fnunirn 5966* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
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Theorem | elunirnALT 5967* | Membership in the union of the range of a function, proved directly. Unlike elunirn 5965, it doesn't appeal to ndmfv 5722 (via funiunfv 5962). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | abrexex2 5968* |
Existence of an existentially restricted class abstraction. ![]() ![]() ![]() |
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Theorem | abexssex 5969* |
Existence of a class abstraction with an existentially quantified
expression. Both ![]() ![]() ![]() |
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Theorem | abexex 5970* | A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.) |
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Theorem | dff13 5971* | 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.) |
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Theorem | f1veqaeq 5972 | 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.) |
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Theorem | dff13f 5973* | 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.) |
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Theorem | f1mpt 5974* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
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Theorem | f1fveq 5975 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
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Theorem | f1elima 5976 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | f1imass 5977 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | f1imaeq 5978 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | f1imapss 5979 | Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | dff1o6 5980* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
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Theorem | f1ocnvfv1 5981 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
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Theorem | f1ocnvfv2 5982 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
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Theorem | f1ocnvfv 5983 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.) |
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Theorem | f1ocnvfvb 5984 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.) |
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Theorem | f1ocnvdm 5985 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
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Theorem | f1ocnvfvrneq 5986 | 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.) |
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Theorem | fcof1 5987 | 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.) |
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Theorem | fcofo 5988 | 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.) |
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Theorem | cbvfo 5989* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
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Theorem | cbvexfo 5990* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
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Theorem | cocan1 5991 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
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Theorem | cocan2 5992 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
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Theorem | fcof1o 5993 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
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Theorem | foeqcnvco 5994 | 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.) |
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Theorem | f1eqcocnv 5995 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
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Theorem | fveqf1o 5996 |
Given a bijection ![]() ![]() |
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Theorem | fliftrel 5997* |
![]() ![]() ![]() ![]() |
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Theorem | fliftel 5998* |
Elementhood in the relation ![]() |
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Theorem | fliftel1 5999* |
Elementhood in the relation ![]() |
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Theorem | fliftcnv 6000* |
Converse of the relation ![]() |
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