Theorem dffo2 6800

Description: Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)

Assertion

Ref	Expression
dffo2	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))

Proof of Theorem dffo2

Step	Hyp	Ref	Expression
1		fof 6796	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		forn 6799	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
3	1, 2	jca 521	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
4		ffn 6709	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		df-fo 6544	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6	5	biimpri 231	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵)
7	4, 6	sylan 592	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵)
8	3, 7	impbii 212	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ran crn 5651   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-ss 3896  df-f 6542  df-fo 6544

This theorem is used by:  focofo  6809  foconst  6811  dff1o5  6834  dffo3  7102  dffo4  7103  exfo  7105  dffo3f  7106  fo1stres  8027  fo2ndres  8028  fo2ndf  8132  cantnf  9694  hsmexlem2  10505  setcepi  18263  odf1o1  19786  efgsfo  19953  pjfo  22021  xrhmeo  25267  grpofo  31101  cnpconn  36037  lnmepi  44130  imasetpreimafvbijlemfo  48530  fargshiftfo  48567