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Theorem dffo2 6761
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 6757 . . 3 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 forn 6760 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
31, 2jca 513 . 2 (𝐹:𝐴onto𝐵 → (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
4 ffn 6669 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 df-fo 6503 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
65biimpri 227 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴onto𝐵)
74, 6sylan 581 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴onto𝐵)
83, 7impbii 208 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  ran crn 5635   Fn wfn 6492  wf 6493  ontowfo 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-in 3918  df-ss 3928  df-f 6501  df-fo 6503
This theorem is referenced by:  focofo  6770  foconst  6772  dff1o5  6794  dffo3  7053  dffo4  7054  exfo  7056  fo1stres  7948  fo2ndres  7949  fo2ndf  8054  cantnf  9630  hsmexlem2  10364  setcepi  17975  odf1o1  19355  efgsfo  19522  pjfo  21124  xrhmeo  24312  grpofo  29444  cnpconn  33827  lnmepi  41415  dffo3f  43405  imasetpreimafvbijlemfo  45604  fargshiftfo  45641
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