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Theorem dffo2 5319
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 5315 . . 3 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 forn 5318 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
31, 2jca 304 . 2 (𝐹:𝐴onto𝐵 → (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
4 ffn 5242 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 df-fo 5099 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
65biimpri 132 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴onto𝐵)
74, 6sylan 281 . 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 1316  ran crn 4510   Fn wfn 5088  wf 5089  ontowfo 5091
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054  df-f 5097  df-fo 5099
This theorem is referenced by:  foco  5325  dff1o5  5344  dffo3  5535  dffo4  5536  fo1stresm  6027  fo2ndresm  6028  fo2ndf  6092  1fv  9884
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