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Theorem dffo2 5317
Description: Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
dffo2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))

Proof of Theorem dffo2
StepHypRef Expression
1 fof 5313 . . 3 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 forn 5316 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
31, 2jca 302 . 2 (𝐹:𝐴onto𝐵 → (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
4 ffn 5240 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 df-fo 5097 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
65biimpri 132 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴onto𝐵)
74, 6sylan 279 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴onto𝐵)
83, 7impbii 125 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1314  ran crn 4508   Fn wfn 5086  wf 5087  ontowfo 5089
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-ss 3052  df-f 5095  df-fo 5097
This theorem is referenced by:  foco  5323  dff1o5  5342  dffo3  5533  dffo4  5534  fo1stresm  6025  fo2ndresm  6026  fo2ndf  6090  1fv  9867
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