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Theorem dffn4 5223
Description: A function maps onto its range. (Contributed by NM, 10-May-1998.)
Assertion
Ref Expression
dffn4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)

Proof of Theorem dffn4
StepHypRef Expression
1 eqid 2088 . . 3 ran 𝐹 = ran 𝐹
21biantru 296 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹))
3 df-fo 5008 . 2 (𝐹:𝐴onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹))
42, 3bitr4i 185 1 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  ran crn 4429   Fn wfn 4997  ontowfo 5000
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-gen 1383  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-fo 5008
This theorem is referenced by:  funforn  5224  ffoss  5269  tposf2  6015  mapsn  6427
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