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Mirrors > Home > MPE Home > Th. List > ffrnbd | Structured version Visualization version GIF version |
Description: A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6683. (Contributed by AV, 20-Sep-2024.) |
Ref | Expression |
---|---|
ffrnbd.r | β’ (π β ran πΉ β π΅) |
Ref | Expression |
---|---|
ffrnbd | β’ (π β (πΉ:π΄βΆπ΅ β πΉ:π΄βΆran πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffrnb 6684 | . 2 β’ (πΉ:π΄βΆπ΅ β (πΉ:π΄βΆran πΉ β§ ran πΉ β π΅)) | |
2 | ffrnbd.r | . . 3 β’ (π β ran πΉ β π΅) | |
3 | 2 | biantrud 533 | . 2 β’ (π β (πΉ:π΄βΆran πΉ β (πΉ:π΄βΆran πΉ β§ ran πΉ β π΅))) |
4 | 1, 3 | bitr4id 290 | 1 β’ (π β (πΉ:π΄βΆπ΅ β πΉ:π΄βΆran πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β wss 3911 ran crn 5635 βΆwf 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-f 6501 |
This theorem is referenced by: (None) |
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