Theorem dff1o3 6615

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 1093	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
2		dff1o2 6614	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
3		df-fo 6355	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	3	anbi1i 625	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
5	1, 2, 4	3bitr4i 305	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ^◡ccnv 5548  ran crn 5550  Fun wfun 6343   Fn wfn 6344  –onto→wfo 6347  –1-1-onto→wf1o 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356

This theorem is referenced by:  f1ofo  6616  resdif  6629  f1opw  7395  f11o  7642  1stconst  7789  2ndconst  7790  curry1  7793  curry2  7796  f1o2ndf1  7812  ssdomg  8549  phplem4  8693  php3  8697  f1opwfi  8822  cantnfp1lem3  9137  fpwwe2lem6  10051  canthp1lem2  10069  odf1o2  18692  dprdf1o  19148  relogf1o  25144  iseupthf1o  27975  padct  30449  ballotlemfrc  31779  poimirlem1  34887  poimirlem2  34888  poimirlem3  34889  poimirlem4  34890  poimirlem6  34892  poimirlem7  34893  poimirlem9  34895  poimirlem11  34897  poimirlem12  34898  poimirlem13  34899  poimirlem14  34900  poimirlem16  34902  poimirlem17  34903  poimirlem19  34905  poimirlem20  34906  poimirlem23  34909  poimirlem24  34910  poimirlem25  34911  poimirlem29  34915  poimirlem31  34917  ntrneifv2  40423