Theorem dff1o3 6809

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:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))

Proof of Theorem dff1o3

Step	Hyp	Ref	Expression
1		3anan32 1096	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
2		dff1o2 6808	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
3		df-fo 6520	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	3	anbi1i 624	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
5	1, 2, 4	3bitr4i 303	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ^◡ccnv 5640  ran crn 5642  Fun wfun 6508   Fn wfn 6509  –onto→wfo 6512  –1-1-onto→wf1o 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-cleq 2722  df-ss 3934  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521

This theorem is referenced by:  f1ofo  6810  resdif  6824  f1opw  7648  f11o  7928  1stconst  8082  2ndconst  8083  curry1  8086  curry2  8089  f1o2ndf1  8104  ssdomg  8974  dif1enlem  9126  dif1enlemOLD  9127  phplem2  9175  php3  9179  f1opwfi  9314  cantnfp1lem3  9640  fpwwe2lem5  10595  canthp1lem2  10613  odf1o2  19510  dprdf1o  19971  relogf1o  26482  iseupthf1o  30138  padct  32650  ballotlemfrc  34525  poimirlem1  37622  poimirlem2  37623  poimirlem3  37624  poimirlem4  37625  poimirlem6  37627  poimirlem7  37628  poimirlem9  37630  poimirlem11  37632  poimirlem12  37633  poimirlem13  37634  poimirlem14  37635  poimirlem16  37637  poimirlem17  37638  poimirlem19  37640  poimirlem20  37641  poimirlem23  37644  poimirlem24  37645  poimirlem25  37646  poimirlem29  37650  poimirlem31  37652  ntrneifv2  44076  permaxpow  45006  upgrimpthslem1  47911  upgrimspths  47914  idfth  49151  idsubc  49153