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Mirrors > Home > MPE Home > Th. List > ffrn | Structured version Visualization version GIF version |
Description: A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
ffrn | ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6507 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | dffn3 6518 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) | |
3 | 1, 2 | sylib 219 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ran crn 5549 Fn wfn 6343 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-in 3940 df-ss 3949 df-f 6352 |
This theorem is referenced by: fo2ndf 7806 mapsnd 8438 itg1val2 24212 selvval2lem4 39014 volicoff 42157 fundcmpsurbijinjpreimafv 43444 |
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