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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | foeq1 6801 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
β’ (πΉ = πΊ β (πΉ:π΄βontoβπ΅ β πΊ:π΄βontoβπ΅)) | ||
Theorem | foeq2 6802 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
β’ (π΄ = π΅ β (πΉ:π΄βontoβπΆ β πΉ:π΅βontoβπΆ)) | ||
Theorem | foeq3 6803 | Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
β’ (π΄ = π΅ β (πΉ:πΆβontoβπ΄ β πΉ:πΆβontoβπ΅)) | ||
Theorem | nffo 6804 | Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.) |
β’ β²π₯πΉ & β’ β²π₯π΄ & β’ β²π₯π΅ β β’ β²π₯ πΉ:π΄βontoβπ΅ | ||
Theorem | fof 6805 | An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.) |
β’ (πΉ:π΄βontoβπ΅ β πΉ:π΄βΆπ΅) | ||
Theorem | fofun 6806 | An onto mapping is a function. (Contributed by NM, 29-Mar-2008.) |
β’ (πΉ:π΄βontoβπ΅ β Fun πΉ) | ||
Theorem | fofn 6807 | An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.) |
β’ (πΉ:π΄βontoβπ΅ β πΉ Fn π΄) | ||
Theorem | forn 6808 | The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.) |
β’ (πΉ:π΄βontoβπ΅ β ran πΉ = π΅) | ||
Theorem | dffo2 6809 | Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.) |
β’ (πΉ:π΄βontoβπ΅ β (πΉ:π΄βΆπ΅ β§ ran πΉ = π΅)) | ||
Theorem | foima 6810 | The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.) |
β’ (πΉ:π΄βontoβπ΅ β (πΉ β π΄) = π΅) | ||
Theorem | dffn4 6811 | A function maps onto its range. (Contributed by NM, 10-May-1998.) |
β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) | ||
Theorem | funforn 6812 | A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.) |
β’ (Fun π΄ β π΄:dom π΄βontoβran π΄) | ||
Theorem | fodmrnu 6813 | An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.) |
β’ ((πΉ:π΄βontoβπ΅ β§ πΉ:πΆβontoβπ·) β (π΄ = πΆ β§ π΅ = π·)) | ||
Theorem | fimadmfo 6814 | A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) |
β’ (πΉ:π΄βΆπ΅ β πΉ:π΄βontoβ(πΉ β π΄)) | ||
Theorem | fores 6815 | Restriction of an onto function. (Contributed by NM, 4-Mar-1997.) |
β’ ((Fun πΉ β§ π΄ β dom πΉ) β (πΉ βΎ π΄):π΄βontoβ(πΉ β π΄)) | ||
Theorem | fimadmfoALT 6816 | Alternate proof of fimadmfo 6814, based on fores 6815. A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (πΉ:π΄βΆπ΅ β πΉ:π΄βontoβ(πΉ β π΄)) | ||
Theorem | focnvimacdmdm 6817 | The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.) |
β’ (πΊ:π΄βontoβπ΅ β (β‘πΊ β π΅) = π΄) | ||
Theorem | focofo 6818 | Composition of onto functions. Generalisation of foco 6819. (Contributed by AV, 29-Sep-2024.) |
β’ ((πΉ:π΄βontoβπ΅ β§ Fun πΊ β§ π΄ β ran πΊ) β (πΉ β πΊ):(β‘πΊ β π΄)βontoβπ΅) | ||
Theorem | foco 6819 | Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.) |
β’ ((πΉ:π΅βontoβπΆ β§ πΊ:π΄βontoβπ΅) β (πΉ β πΊ):π΄βontoβπΆ) | ||
Theorem | foconst 6820 | A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.) |
β’ ((πΉ:π΄βΆ{π΅} β§ πΉ β β ) β πΉ:π΄βontoβ{π΅}) | ||
Theorem | f1oeq1 6821 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
β’ (πΉ = πΊ β (πΉ:π΄β1-1-ontoβπ΅ β πΊ:π΄β1-1-ontoβπ΅)) | ||
Theorem | f1oeq2 6822 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
β’ (π΄ = π΅ β (πΉ:π΄β1-1-ontoβπΆ β πΉ:π΅β1-1-ontoβπΆ)) | ||
Theorem | f1oeq3 6823 | Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
β’ (π΄ = π΅ β (πΉ:πΆβ1-1-ontoβπ΄ β πΉ:πΆβ1-1-ontoβπ΅)) | ||
Theorem | f1oeq23 6824 | Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.) |
β’ ((π΄ = π΅ β§ πΆ = π·) β (πΉ:π΄β1-1-ontoβπΆ β πΉ:π΅β1-1-ontoβπ·)) | ||
Theorem | f1eq123d 6825 | Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (πΉ:π΄β1-1βπΆ β πΊ:π΅β1-1βπ·)) | ||
Theorem | foeq123d 6826 | Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (πΉ:π΄βontoβπΆ β πΊ:π΅βontoβπ·)) | ||
Theorem | f1oeq123d 6827 | Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β (πΉ:π΄β1-1-ontoβπΆ β πΊ:π΅β1-1-ontoβπ·)) | ||
Theorem | f1oeq1d 6828 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β πΉ = πΊ) β β’ (π β (πΉ:π΄β1-1-ontoβπ΅ β πΊ:π΄β1-1-ontoβπ΅)) | ||
Theorem | f1oeq2d 6829 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β π΄ = π΅) β β’ (π β (πΉ:π΄β1-1-ontoβπΆ β πΉ:π΅β1-1-ontoβπΆ)) | ||
Theorem | f1oeq3d 6830 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
β’ (π β π΄ = π΅) β β’ (π β (πΉ:πΆβ1-1-ontoβπ΄ β πΉ:πΆβ1-1-ontoβπ΅)) | ||
Theorem | nff1o 6831 | Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.) |
β’ β²π₯πΉ & β’ β²π₯π΄ & β’ β²π₯π΅ β β’ β²π₯ πΉ:π΄β1-1-ontoβπ΅ | ||
Theorem | f1of1 6832 | A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ:π΄β1-1βπ΅) | ||
Theorem | f1of 6833 | A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ:π΄βΆπ΅) | ||
Theorem | f1ofn 6834 | A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ Fn π΄) | ||
Theorem | f1ofun 6835 | A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β Fun πΉ) | ||
Theorem | f1orel 6836 | A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β Rel πΉ) | ||
Theorem | f1odm 6837 | The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β dom πΉ = π΄) | ||
Theorem | dff1o2 6838 | Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = π΅)) | ||
Theorem | dff1o3 6839 | Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ:π΄βontoβπ΅ β§ Fun β‘πΉ)) | ||
Theorem | f1ofo 6840 | A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β πΉ:π΄βontoβπ΅) | ||
Theorem | dff1o4 6841 | Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ Fn π΄ β§ β‘πΉ Fn π΅)) | ||
Theorem | dff1o5 6842 | Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ:π΄β1-1βπ΅ β§ ran πΉ = π΅)) | ||
Theorem | f1orn 6843 | A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ)) | ||
Theorem | f1f1orn 6844 | A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.) |
β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄β1-1-ontoβran πΉ) | ||
Theorem | f1ocnv 6845 | The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β β‘πΉ:π΅β1-1-ontoβπ΄) | ||
Theorem | f1ocnvb 6846 | A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/range interchanged. (Contributed by NM, 8-Dec-2003.) |
β’ (Rel πΉ β (πΉ:π΄β1-1-ontoβπ΅ β β‘πΉ:π΅β1-1-ontoβπ΄)) | ||
Theorem | f1ores 6847 | The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1-ontoβ(πΉ β πΆ)) | ||
Theorem | f1orescnv 6848 | The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.) |
β’ ((Fun β‘πΉ β§ (πΉ βΎ π ):π β1-1-ontoβπ) β (β‘πΉ βΎ π):πβ1-1-ontoβπ ) | ||
Theorem | f1imacnv 6849 | Preimage of an image. (Contributed by NM, 30-Sep-2004.) |
β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (β‘πΉ β (πΉ β πΆ)) = πΆ) | ||
Theorem | foimacnv 6850 | A reverse version of f1imacnv 6849. (Contributed by Jeff Hankins, 16-Jul-2009.) |
β’ ((πΉ:π΄βontoβπ΅ β§ πΆ β π΅) β (πΉ β (β‘πΉ β πΆ)) = πΆ) | ||
Theorem | foun 6851 | The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.) |
β’ (((πΉ:π΄βontoβπ΅ β§ πΊ:πΆβontoβπ·) β§ (π΄ β© πΆ) = β ) β (πΉ βͺ πΊ):(π΄ βͺ πΆ)βontoβ(π΅ βͺ π·)) | ||
Theorem | f1oun 6852 | The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.) |
β’ (((πΉ:π΄β1-1-ontoβπ΅ β§ πΊ:πΆβ1-1-ontoβπ·) β§ ((π΄ β© πΆ) = β β§ (π΅ β© π·) = β )) β (πΉ βͺ πΊ):(π΄ βͺ πΆ)β1-1-ontoβ(π΅ βͺ π·)) | ||
Theorem | f1un 6853 | The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.) |
β’ (((πΉ:π΄β1-1βπ΅ β§ πΊ:πΆβ1-1βπ·) β§ ((π΄ β© πΆ) = β β§ (π΅ β© π·) = β )) β (πΉ βͺ πΊ):(π΄ βͺ πΆ)β1-1β(π΅ βͺ π·)) | ||
Theorem | resdif 6854 | The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.) |
β’ ((Fun β‘πΉ β§ (πΉ βΎ π΄):π΄βontoβπΆ β§ (πΉ βΎ π΅):π΅βontoβπ·) β (πΉ βΎ (π΄ β π΅)):(π΄ β π΅)β1-1-ontoβ(πΆ β π·)) | ||
Theorem | resin 6855 | The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.) |
β’ ((Fun β‘πΉ β§ (πΉ βΎ π΄):π΄βontoβπΆ β§ (πΉ βΎ π΅):π΅βontoβπ·) β (πΉ βΎ (π΄ β© π΅)):(π΄ β© π΅)β1-1-ontoβ(πΆ β© π·)) | ||
Theorem | f1oco 6856 | Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.) |
β’ ((πΉ:π΅β1-1-ontoβπΆ β§ πΊ:π΄β1-1-ontoβπ΅) β (πΉ β πΊ):π΄β1-1-ontoβπΆ) | ||
Theorem | f1cnv 6857 | The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.) |
β’ (πΉ:π΄β1-1βπ΅ β β‘πΉ:ran πΉβ1-1-ontoβπ΄) | ||
Theorem | funcocnv2 6858 | Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
β’ (Fun πΉ β (πΉ β β‘πΉ) = ( I βΎ ran πΉ)) | ||
Theorem | fococnv2 6859 | The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
β’ (πΉ:π΄βontoβπ΅ β (πΉ β β‘πΉ) = ( I βΎ π΅)) | ||
Theorem | f1ococnv2 6860 | The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (πΉ β β‘πΉ) = ( I βΎ π΅)) | ||
Theorem | f1cocnv2 6861 | Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.) |
β’ (πΉ:π΄β1-1βπ΅ β (πΉ β β‘πΉ) = ( I βΎ ran πΉ)) | ||
Theorem | f1ococnv1 6862 | The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.) |
β’ (πΉ:π΄β1-1-ontoβπ΅ β (β‘πΉ β πΉ) = ( I βΎ π΄)) | ||
Theorem | f1cocnv1 6863 | Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.) |
β’ (πΉ:π΄β1-1βπ΅ β (β‘πΉ β πΉ) = ( I βΎ π΄)) | ||
Theorem | funcoeqres 6864 | Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
β’ ((Fun πΊ β§ (πΉ β πΊ) = π») β (πΉ βΎ ran πΊ) = (π» β β‘πΊ)) | ||
Theorem | f1ssf1 6865 | A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.) |
β’ ((Fun πΉ β§ Fun β‘πΉ β§ πΊ β πΉ) β Fun β‘πΊ) | ||
Theorem | f10 6866 | The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.) |
β’ β :β β1-1βπ΄ | ||
Theorem | f10d 6867 | The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.) |
β’ (π β πΉ = β ) β β’ (π β πΉ:dom πΉβ1-1βπ΄) | ||
Theorem | f1o00 6868 | One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.) |
β’ (πΉ:β β1-1-ontoβπ΄ β (πΉ = β β§ π΄ = β )) | ||
Theorem | fo00 6869 | Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.) |
β’ (πΉ:β βontoβπ΄ β (πΉ = β β§ π΄ = β )) | ||
Theorem | f1o0 6870 | One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.) |
β’ β :β β1-1-ontoββ | ||
Theorem | f1oi 6871 | A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ ( I βΎ π΄):π΄β1-1-ontoβπ΄ | ||
Theorem | f1ovi 6872 | The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.) |
β’ I :Vβ1-1-ontoβV | ||
Theorem | f1osn 6873 | A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
β’ π΄ β V & β’ π΅ β V β β’ {β¨π΄, π΅β©}:{π΄}β1-1-ontoβ{π΅} | ||
Theorem | f1osng 6874 | A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
β’ ((π΄ β π β§ π΅ β π) β {β¨π΄, π΅β©}:{π΄}β1-1-ontoβ{π΅}) | ||
Theorem | f1sng 6875 | A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.) |
β’ ((π΄ β π β§ π΅ β π) β {β¨π΄, π΅β©}:{π΄}β1-1βπ) | ||
Theorem | fsnd 6876 | A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π) β β’ (π β {β¨π΄, π΅β©}:{π΄}βΆπ) | ||
Theorem | f1oprswap 6877 | A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ ((π΄ β π β§ π΅ β π) β {β¨π΄, π΅β©, β¨π΅, π΄β©}:{π΄, π΅}β1-1-ontoβ{π΄, π΅}) | ||
Theorem | f1oprg 6878 | An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6877. (Contributed by Alexander van der Vekens, 14-Aug-2017.) |
β’ (((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β ((π΄ β πΆ β§ π΅ β π·) β {β¨π΄, π΅β©, β¨πΆ, π·β©}:{π΄, πΆ}β1-1-ontoβ{π΅, π·})) | ||
Theorem | tz6.12-2 6879* | Function value when πΉ is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
β’ (Β¬ β!π₯ π΄πΉπ₯ β (πΉβπ΄) = β ) | ||
Theorem | fveu 6880* | The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.) |
β’ (β!π₯ π΄πΉπ₯ β (πΉβπ΄) = βͺ {π₯ β£ π΄πΉπ₯}) | ||
Theorem | brprcneu 6881* | If π΄ is a proper class and πΉ is any class, then there is no unique set which is related to π΄ through the binary relation πΉ. See brprcneuALT 6882 for a proof that uses ax-pow 5363 instead of ax-pr 5427. (Contributed by Scott Fenton, 7-Oct-2017.) |
β’ (Β¬ π΄ β V β Β¬ β!π₯ π΄πΉπ₯) | ||
Theorem | brprcneuALT 6882* | Alternate proof of brprcneu 6881 using ax-pow 5363 instead of ax-pr 5427. (Contributed by Scott Fenton, 7-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (Β¬ π΄ β V β Β¬ β!π₯ π΄πΉπ₯) | ||
Theorem | fvprc 6883 | A function's value at a proper class is the empty set. See fvprcALT 6884 for a proof that uses ax-pow 5363 instead of ax-pr 5427. (Contributed by NM, 20-May-1998.) Avoid ax-pow 5363. (Revised by BTernaryTau, 3-Aug-2024.) (Proof shortened by BTernaryTau, 3-Dec-2024.) |
β’ (Β¬ π΄ β V β (πΉβπ΄) = β ) | ||
Theorem | fvprcALT 6884 | Alternate proof of fvprc 6883 using ax-pow 5363 instead of ax-pr 5427. (Contributed by NM, 20-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (Β¬ π΄ β V β (πΉβπ΄) = β ) | ||
Theorem | rnfvprc 6885 | The range of a function value at a proper class is empty. (Contributed by AV, 20-Aug-2022.) |
β’ π = (πΉβπ) β β’ (Β¬ π β V β ran π = β ) | ||
Theorem | fv2 6886* | Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
β’ (πΉβπ΄) = βͺ {π₯ β£ βπ¦(π΄πΉπ¦ β π¦ = π₯)} | ||
Theorem | dffv3 6887* | A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.) |
β’ (πΉβπ΄) = (β©π₯π₯ β (πΉ β {π΄})) | ||
Theorem | dffv4 6888* | The previous definition of function value, from before the β© operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 6091), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.) |
β’ (πΉβπ΄) = βͺ {π₯ β£ (πΉ β {π΄}) = {π₯}} | ||
Theorem | elfv 6889* | Membership in a function value. (Contributed by NM, 30-Apr-2004.) |
β’ (π΄ β (πΉβπ΅) β βπ₯(π΄ β π₯ β§ βπ¦(π΅πΉπ¦ β π¦ = π₯))) | ||
Theorem | fveq1 6890 | Equality theorem for function value. (Contributed by NM, 29-Dec-1996.) |
β’ (πΉ = πΊ β (πΉβπ΄) = (πΊβπ΄)) | ||
Theorem | fveq2 6891 | Equality theorem for function value. (Contributed by NM, 29-Dec-1996.) |
β’ (π΄ = π΅ β (πΉβπ΄) = (πΉβπ΅)) | ||
Theorem | fveq1i 6892 | Equality inference for function value. (Contributed by NM, 2-Sep-2003.) |
β’ πΉ = πΊ β β’ (πΉβπ΄) = (πΊβπ΄) | ||
Theorem | fveq1d 6893 | Equality deduction for function value. (Contributed by NM, 2-Sep-2003.) |
β’ (π β πΉ = πΊ) β β’ (π β (πΉβπ΄) = (πΊβπ΄)) | ||
Theorem | fveq2i 6894 | Equality inference for function value. (Contributed by NM, 28-Jul-1999.) |
β’ π΄ = π΅ β β’ (πΉβπ΄) = (πΉβπ΅) | ||
Theorem | fveq2d 6895 | Equality deduction for function value. (Contributed by NM, 29-May-1999.) |
β’ (π β π΄ = π΅) β β’ (π β (πΉβπ΄) = (πΉβπ΅)) | ||
Theorem | 2fveq3 6896 | Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.) |
β’ (π΄ = π΅ β (πΉβ(πΊβπ΄)) = (πΉβ(πΊβπ΅))) | ||
Theorem | fveq12i 6897 | Equality deduction for function value. (Contributed by FL, 27-Jun-2014.) |
β’ πΉ = πΊ & β’ π΄ = π΅ β β’ (πΉβπ΄) = (πΊβπ΅) | ||
Theorem | fveq12d 6898 | Equality deduction for function value. (Contributed by FL, 22-Dec-2008.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) β β’ (π β (πΉβπ΄) = (πΊβπ΅)) | ||
Theorem | fveqeq2d 6899 | Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.) |
β’ (π β π΄ = π΅) β β’ (π β ((πΉβπ΄) = πΆ β (πΉβπ΅) = πΆ)) | ||
Theorem | fveqeq2 6900 | Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.) |
β’ (π΄ = π΅ β ((πΉβπ΄) = πΆ β (πΉβπ΅) = πΆ)) |
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