Theorem dff1o3 6181

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 1068	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
2		dff1o2 6180	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
3		df-fo 5932	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	3	anbi1i 731	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
5	1, 2, 4	3bitr4i 292	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ^◡ccnv 5142  ran crn 5144  Fun wfun 5920   Fn wfn 5921  –onto→wfo 5924  –1-1-onto→wf1o 5925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933

This theorem is referenced by:  f1ofo  6182  resdif  6195  f1opw  6931  f11o  7170  1stconst  7310  2ndconst  7311  curry1  7314  curry2  7317  f1o2ndf1  7330  ssdomg  8043  phplem4  8183  php3  8187  f1opwfi  8311  cantnfp1lem3  8615  fpwwe2lem6  9495  canthp1lem2  9513  odf1o2  18034  dprdf1o  18477  relogf1o  24358  iseupthf1o  27180  padct  29625  ballotlemfrc  30716  poimirlem1  33540  poimirlem2  33541  poimirlem3  33542  poimirlem4  33543  poimirlem6  33545  poimirlem7  33546  poimirlem9  33548  poimirlem11  33550  poimirlem12  33551  poimirlem13  33552  poimirlem14  33553  poimirlem16  33555  poimirlem17  33556  poimirlem19  33558  poimirlem20  33559  poimirlem23  33562  poimirlem24  33563  poimirlem25  33564  poimirlem29  33568  poimirlem31  33570  ntrneifv2  38695