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Theorem dffn3 5289
Description: A function maps to its range. (Contributed by NM, 1-Sep-1999.)
Assertion
Ref Expression
dffn3 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)

Proof of Theorem dffn3
StepHypRef Expression
1 ssid 3120 . . 3 ran 𝐹 ⊆ ran 𝐹
21biantru 300 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
3 df-f 5133 . 2 (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
42, 3bitr4i 186 1 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wss 3074  ran crn 4546   Fn wfn 5124  wf 5125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3080  df-ss 3087  df-f 5133
This theorem is referenced by:  fsn2  5600  fo2ndf  6130
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