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Type | Label | Description |
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Statement | ||
Theorem | fvsn 5301 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.) |
Theorem | fvsng 5302 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) |
Theorem | fvsnun1 5303 | The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5304. (Contributed by NM, 23-Sep-2007.) |
Theorem | fvsnun2 5304 | The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5303. (Contributed by NM, 23-Sep-2007.) |
Theorem | fsnunf 5305 | Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | fsnunfv 5306 | 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 5307 | Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | fvpr1 5308 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Theorem | fvpr2 5309 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Theorem | fvpr1g 5310 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Theorem | fvpr2g 5311 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Theorem | fvtp1g 5312 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp2g 5313 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp3g 5314 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Theorem | fvtp1 5315 | The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvtp2 5316 | The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvtp3 5317 | The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fvconst2g 5318 | The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
Theorem | fconst2g 5319 | A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.) |
Theorem | fvconst2 5320 | The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
Theorem | fconst2 5321 | A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.) |
Theorem | fconstfvm 5322* | A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5321. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Theorem | fconst3m 5323* | Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Theorem | fconst4m 5324* | Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
Theorem | resfunexg 5325 | 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 5326 | 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 5325. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | funex 5327 | 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 5326. (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 5328* | Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
Theorem | mptexg 5329* | 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 5330* | 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 5331 | If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.) |
Theorem | eufnfv 5332* | A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
Theorem | funfvima 5333 | A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
Theorem | funfvima2 5334 | A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
Theorem | funfvima3 5335 | 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 5336 | 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 | rexima 5337* | Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | ralima 5338* | Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | idref 5339* |
TODO: This is the same as issref 4650 (which has a much longer proof).
Should we replace issref 4650 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 5340* | Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Theorem | abrexco 5341* | Composition of two image maps and . (Contributed by NM, 27-May-2013.) |
Theorem | imaiun 5342* | The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Theorem | imauni 5343* | 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 5344* | 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 5345* | 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 5344. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | funiunfvdmf 5346* | The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5345 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | eluniimadm 5347* | Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | elunirn 5348* | Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.) |
Theorem | fnunirn 5349* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Theorem | dff13 5350* | 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 5351 | 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 5352* | 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 5353* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | f1fveq 5354 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
Theorem | f1elima 5355 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | f1imass 5356 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | f1imaeq 5357 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | f1imapss 5358 | Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | dff1o6 5359* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
Theorem | f1ocnvfv1 5360 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv2 5361 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv 5362 | 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 5363 | 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 5364 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfvrneq 5365 | 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 5366 | 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 5367 | 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 5368* | 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 5369* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
Theorem | cocan1 5370 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Theorem | cocan2 5371 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Theorem | fcof1o 5372 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Theorem | foeqcnvco 5373 | 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 5374 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Theorem | fliftrel 5375* | , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel 5376* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel1 5377* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftcnv 5378* | Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfun 5379* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfund 5380* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfuns 5381* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftf 5382* | The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftval 5383* | The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | isoeq1 5384 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq2 5385 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq3 5386 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq4 5387 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq5 5388 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | nfiso 5389 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | isof1o 5390 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
Theorem | isorel 5391 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
Theorem | isoresbr 5392* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Theorem | isoid 5393 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv 5394 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv2 5395 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
Theorem | isores2 5396 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores1 5397 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores3 5398 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
Theorem | isotr 5399 | 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 | isoini 5400 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
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