ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dff1o2 Structured version   Unicode version

Theorem dff1o2 5074
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o2  F : -1-1-onto->  F  Fn  Fun  `' F  ran  F

Proof of Theorem dff1o2
StepHypRef Expression
1 df-f1o 4852 . 2  F : -1-1-onto->  F : -1-1->  F : -onto->
2 df-f1 4850 . . . 4  F : -1-1->  F : -->  Fun  `' F
3 df-fo 4851 . . . 4  F : -onto->  F  Fn  ran  F
42, 3anbi12i 433 . . 3  F : -1-1->  F : -onto->  F :
-->  Fun  `' F  F  Fn  ran  F
5 anass 381 . . . 4  F : -->  Fun  `' F  F  Fn  ran  F  F : -->  Fun  `' F  F  Fn  ran  F
6 3anan12 896 . . . . . 6  F  Fn  Fun  `' F  ran  F  Fun  `' F  F  Fn  ran  F
76anbi1i 431 . . . . 5  F  Fn  Fun  `' F  ran  F  F : -->  Fun  `' F  F  Fn  ran  F  F : -->
8 eqimss 2991 . . . . . . . 8  ran 
F  ran 
F  C_
9 df-f 4849 . . . . . . . . 9  F : -->  F  Fn  ran  F 
C_
109biimpri 124 . . . . . . . 8  F  Fn  ran  F  C_  F : -->
118, 10sylan2 270 . . . . . . 7  F  Fn  ran  F  F : -->
12113adant2 922 . . . . . 6  F  Fn  Fun  `' F  ran  F  F : -->
1312pm4.71i 371 . . . . 5  F  Fn  Fun  `' F  ran  F  F  Fn  Fun  `' F  ran  F  F : -->
14 ancom 253 . . . . 5  F : -->  Fun  `' F  F  Fn  ran  F  Fun  `' F  F  Fn  ran  F  F :
-->
157, 13, 143bitr4ri 202 . . . 4  F : -->  Fun  `' F  F  Fn  ran  F  F  Fn  Fun  `' F  ran  F
165, 15bitri 173 . . 3  F : -->  Fun  `' F  F  Fn  ran  F  F  Fn  Fun  `' F  ran  F
174, 16bitri 173 . 2  F : -1-1->  F : -onto->  F  Fn  Fun  `' F  ran  F
181, 17bitri 173 1  F : -1-1-onto->  F  Fn  Fun  `' F  ran  F
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   w3a 884   wceq 1242    C_ wss 2911   `'ccnv 4287   ran crn 4289   Fun wfun 4839    Fn wfn 4840   -->wf 4841   -1-1->wf1 4842   -onto->wfo 4843   -1-1-onto->wf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  dff1o3  5075  dff1o4  5077  f1orn  5079
  Copyright terms: Public domain W3C validator