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Theorem dffn3 5171
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 3044 . . 3 ran 𝐹 ⊆ ran 𝐹
21biantru 296 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
3 df-f 5019 . 2 (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
42, 3bitr4i 185 1 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wss 2999  ran crn 4439   Fn wfn 5010  wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012  df-f 5019
This theorem is referenced by:  fsn2  5471  fo2ndf  5992
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