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Theorem dffn4 5139
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 2056 . . 3 ran 𝐹 = ran 𝐹
21biantru 290 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹))
3 df-fo 4935 . 2 (𝐹:𝐴onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹))
42, 3bitr4i 180 1 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  ran crn 4373   Fn wfn 4924  ontowfo 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-fo 4935
This theorem is referenced by:  funforn  5140  ffoss  5185  tposf2  5913
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