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

Theorem dff1o5 5385
 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 5139 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 dffo2 5358 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
3 f1f 5337 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
43biantrurd 303 . . . 4 (𝐹:𝐴1-1𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵)))
52, 4bitr4id 198 . . 3 (𝐹:𝐴1-1𝐵 → (𝐹:𝐴onto𝐵 ↔ ran 𝐹 = 𝐵))
65pm5.32i 450 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
71, 6bitri 183 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332  ran crn 4549  ⟶wf 5128  –1-1→wf1 5129  –onto→wfo 5130  –1-1-onto→wf1o 5131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3083  df-ss 3090  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139 This theorem is referenced by:  f1orescnv  5392  f1finf1o  6845  djuinr  6958  eninl  6992  eninr  6993  frec2uzf1od  10230  ennnfonelemex  11983  ennnfonelemen  11990  pwf1oexmid  13386
 Copyright terms: Public domain W3C validator