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Type | Label | Description |
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Statement | ||
Theorem | fnex 5501 | 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 5500. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | funex 5502 | 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 5501. (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 5503* | Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
Theorem | mptexg 5504* | 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 5505* | 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 5506 | If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.) |
Theorem | eufnfv 5507* | A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
Theorem | funfvima 5508 | A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
Theorem | funfvima2 5509 | A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
Theorem | funfvima3 5510 | 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 5511 | 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 5512* | 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 5222). (Contributed by BJ, 6-Jul-2022.) |
Theorem | foelrn 5513* | Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.) |
Theorem | foelrnOLD 5514* | Obsolete proof of foelrn 5513 as of 6-Jul-2022. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | foco2 5515 | 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 5516* | Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | ralima 5517* | Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Theorem | idref 5518* |
TODO: This is the same as issref 4801 (which has a much longer proof).
Should we replace issref 4801 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 5519* | Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Theorem | abrexco 5520* | Composition of two image maps and . (Contributed by NM, 27-May-2013.) |
Theorem | imaiun 5521* | The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Theorem | imauni 5522* | 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 5523* | 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 5524* | 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 5523. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | funiunfvdmf 5525* | The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5524 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | eluniimadm 5526* | Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Theorem | elunirn 5527* | Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.) |
Theorem | fnunirn 5528* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Theorem | dff13 5529* | 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 5530 | 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 5531* | 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 5532* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | f1fveq 5533 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
Theorem | f1elima 5534 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | f1imass 5535 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | f1imaeq 5536 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Theorem | dff1o6 5537* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
Theorem | f1ocnvfv1 5538 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv2 5539 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Theorem | f1ocnvfv 5540 | 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 5541 | 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 5542 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
Theorem | f1ocnvfvrneq 5543 | 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 5544 | 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 5545 | 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 5546* | 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 5547* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
Theorem | cocan1 5548 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Theorem | cocan2 5549 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Theorem | fcof1o 5550 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Theorem | foeqcnvco 5551 | 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 5552 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Theorem | fliftrel 5553* | , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel 5554* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftel1 5555* | Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftcnv 5556* | Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfun 5557* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfund 5558* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftfuns 5559* | The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftf 5560* | The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | fliftval 5561* | The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | isoeq1 5562 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq2 5563 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq3 5564 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq4 5565 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | isoeq5 5566 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Theorem | nfiso 5567 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | isof1o 5568 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
Theorem | isorel 5569 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
Theorem | isoresbr 5570* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Theorem | isoid 5571 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv 5572 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Theorem | isocnv2 5573 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
Theorem | isores2 5574 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores1 5575 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | isores3 5576 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
Theorem | isotr 5577 | 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 5578 | The empty set is an isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
Theorem | isoini 5579 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
Theorem | isoini2 5580 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
Theorem | isoselem 5581* | Lemma for isose 5582. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isose 5582 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Se Se | ||
Theorem | isopolem 5583 | Lemma for isopo 5584. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isopo 5584 | An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isosolem 5585 | Lemma for isoso 5586. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | isoso 5586 | An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Theorem | f1oiso 5587* | Any one-to-one onto function determines an isomorphism with an induced relation . Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.) |
Theorem | f1oiso2 5588* | Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.) |
Syntax | crio 5589 | Extend class notation with restricted description binder. |
Definition | df-riota 5590 | Define restricted description binder. In case there is no unique such that holds, it evaluates to the empty set. See also comments for df-iota 4967. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.) |
Theorem | riotaeqdv 5591* | Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotabidv 5592* | Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaeqbidv 5593* | Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.) |
Theorem | riotaexg 5594* | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
Theorem | riotav 5595 | An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
Theorem | riotauni 5596 | Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.) |
Theorem | nfriota1 5597* | The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfriotadxy 5598* | Deduction version of nfriota 5599. (Contributed by Jim Kingdon, 12-Jan-2019.) |
Theorem | nfriota 5599* | A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.) |
Theorem | cbvriota 5600* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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