Theorem dffo2 6596

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 6592	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		forn 6595	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
3	1, 2	jca 514	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
4		ffn 6516	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		df-fo 6363	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6	5	biimpri 230	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵)
7	4, 6	sylan 582	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵)
8	3, 7	impbii 211	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537  ran crn 5558   Fn wfn 6352  ⟶wf 6353  –onto→wfo 6355

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-in 3945  df-ss 3954  df-f 6361  df-fo 6363

This theorem is referenced by:  foco  6604  foconst  6605  dff1o5  6626  dffo3  6870  dffo4  6871  exfo  6873  fo1stres  7717  fo2ndres  7718  fo2ndf  7819  cantnf  9158  hsmexlem2  9851  setcepi  17350  odf1o1  18699  efgsfo  18867  pjfo  20861  xrhmeo  23552  grpofo  28278  cnpconn  32479  lnmepi  39692  dffo3f  41445  imasetpreimafvbijlemfo  43572  fargshiftfo  43609