ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dff1o5 GIF version

Theorem dff1o5 5163
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 4937 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 f1f 5120 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
32biantrurd 293 . . . 4 (𝐹:𝐴1-1𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵)))
4 dffo2 5138 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
53, 4syl6rbbr 192 . . 3 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴onto𝐵 ↔ ran 𝐹 = 𝐵))
65pm5.32i 435 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
71, 6bitri 177 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  ran crn 4374  wf 4926  1-1wf1 4927  ontowfo 4928  1-1-ontowf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  f1orescnv  5170  frec2uzf1od  9356
  Copyright terms: Public domain W3C validator