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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 6722. (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 6538 | . 2 β’ (πΉ:π΄βΆπ΅ β (πΉ Fn π΄ β§ ran πΉ β π΅)) | |
2 | dffn3 6721 | . . 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 3941 ran crn 5668 Fn wfn 6529 βΆwf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3948 df-ss 3958 df-f 6538 |
This theorem is referenced by: ffrnbd 6724 |
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