Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff1o5 Structured version   Visualization version   GIF version

Theorem dff1o5 6059
 Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o5 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))

Proof of Theorem dff1o5
StepHypRef Expression
1 df-f1o 5811 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 f1f 6014 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
32biantrurd 528 . . . 4 (𝐹:𝐴1-1𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵)))
4 dffo2 6032 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
53, 4syl6rbbr 278 . . 3 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴onto𝐵 ↔ ran 𝐹 = 𝐵))
65pm5.32i 667 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
71, 6bitri 263 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  f1orescnv  6065  domdifsn  7928  sucdom2  8041  ackbij1  8943  ackbij2  8948  fin4en1  9014  om2uzf1oi  12614  s4f1o  13513  fvcosymgeq  17672  indlcim  19998  2lgslem1b  24917  ausisusgra  25884  usgraexmpledg  25932  cdleme50f1o  34852  diaf1oN  35437  pwssplit4  36677  meadjiunlem  39358  ausgrusgrb  40395  usgrexmpledg  40486
 Copyright terms: Public domain W3C validator