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Mirrors > Home > MPE Home > Th. List > ffrnb | Structured version Visualization version GIF version |
Description: Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6730. (Contributed by BJ, 21-Sep-2024.) |
Ref | Expression |
---|---|
ffrnb | β’ (πΉ:π΄βΆπ΅ β (πΉ:π΄βΆran πΉ β§ ran πΉ β π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 6546 | . 2 β’ (πΉ:π΄βΆπ΅ β (πΉ Fn π΄ β§ ran πΉ β π΅)) | |
2 | dffn3 6729 | . . 3 β’ (πΉ Fn π΄ β πΉ:π΄βΆran πΉ) | |
3 | 2 | anbi1i 623 | . 2 β’ ((πΉ Fn π΄ β§ ran πΉ β π΅) β (πΉ:π΄βΆran πΉ β§ ran πΉ β π΅)) |
4 | 1, 3 | bitri 275 | 1 β’ (πΉ:π΄βΆπ΅ β (πΉ:π΄βΆran πΉ β§ ran πΉ β π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 β wss 3945 ran crn 5673 Fn wfn 6537 βΆwf 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-in 3952 df-ss 3962 df-f 6546 |
This theorem is referenced by: ffrnbd 6732 |
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