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Theorem dff1o3 6791
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))

Proof of Theorem dff1o3
StepHypRef Expression
1 3anan32 1098 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun 𝐹))
2 dff1o2 6790 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
3 df-fo 6503 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
43anbi1i 625 . 2 ((𝐹:𝐴onto𝐵 ∧ Fun 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun 𝐹))
51, 2, 43bitr4i 303 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088   = wceq 1542  ccnv 5633  ran crn 5635  Fun wfun 6491   Fn wfn 6492  ontowfo 6495  1-1-ontowf1o 6496
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-3an 1090  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-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  f1ofo  6792  resdif  6806  f1opw  7610  f11o  7880  1stconst  8033  2ndconst  8034  curry1  8037  curry2  8040  f1o2ndf1  8055  ssdomg  8941  dif1enlem  9101  dif1enlemOLD  9102  phplem2  9153  php3  9157  phplem4OLD  9165  php3OLD  9169  f1opwfi  9301  cantnfp1lem3  9617  fpwwe2lem5  10572  canthp1lem2  10590  odf1o2  19356  dprdf1o  19812  relogf1o  25925  iseupthf1o  29149  padct  31639  ballotlemfrc  33129  poimirlem1  36082  poimirlem2  36083  poimirlem3  36084  poimirlem4  36085  poimirlem6  36087  poimirlem7  36088  poimirlem9  36090  poimirlem11  36092  poimirlem12  36093  poimirlem13  36094  poimirlem14  36095  poimirlem16  36097  poimirlem17  36098  poimirlem19  36100  poimirlem20  36101  poimirlem23  36104  poimirlem24  36105  poimirlem25  36106  poimirlem29  36110  poimirlem31  36112  ntrneifv2  42359
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