Theorem dffo2 6735

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 6731	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		forn 6734	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
3	1, 2	jca 511	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
4		ffn 6647	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		df-fo 6483	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6	5	biimpri 228	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵)
7	4, 6	sylan 580	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵)
8	3, 7	impbii 209	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ran crn 5615   Fn wfn 6472  ⟶wf 6473  –onto→wfo 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2722  df-ss 3917  df-f 6481  df-fo 6483

This theorem is referenced by:  focofo  6744  foconst  6746  dff1o5  6768  dffo3  7030  dffo4  7031  exfo  7033  dffo3f  7034  fo1stres  7942  fo2ndres  7943  fo2ndf  8046  cantnf  9578  hsmexlem2  10310  setcepi  17987  odf1o1  19477  efgsfo  19644  pjfo  21645  xrhmeo  24864  grpofo  30469  cnpconn  35242  lnmepi  43097  imasetpreimafvbijlemfo  47415  fargshiftfo  47452