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Mirrors > Home > ILE Home > Th. List > dffn4 | GIF version |
Description: A function maps onto its range. (Contributed by NM, 10-May-1998.) |
Ref | Expression |
---|---|
dffn4 | β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 β’ ran πΉ = ran πΉ | |
2 | 1 | biantru 302 | . 2 β’ (πΉ Fn π΄ β (πΉ Fn π΄ β§ ran πΉ = ran πΉ)) |
3 | df-fo 5224 | . 2 β’ (πΉ:π΄βontoβran πΉ β (πΉ Fn π΄ β§ ran πΉ = ran πΉ)) | |
4 | 2, 3 | bitr4i 187 | 1 β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) |
Colors of variables: wff set class |
Syntax hints: β§ wa 104 β wb 105 = wceq 1353 ran crn 4629 Fn wfn 5213 βontoβwfo 5216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-gen 1449 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-fo 5224 |
This theorem is referenced by: funforn 5447 ffoss 5495 tposf2 6271 mapsn 6692 fifo 6981 quslem 12750 |
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