ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dff1o6 Structured version   Unicode version

Theorem dff1o6 5359
Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
Assertion
Ref Expression
dff1o6  F : -1-1-onto->  F  Fn  ran  F  F `
 F `
Distinct variable groups:   ,,   , F,
Allowed substitution hints:   (,)

Proof of Theorem dff1o6
StepHypRef Expression
1 df-f1o 4852 . 2  F : -1-1-onto->  F : -1-1->  F : -onto->
2 dff13 5350 . . 3  F : -1-1->  F : -->  F `  F `
3 df-fo 4851 . . 3  F : -onto->  F  Fn  ran  F
42, 3anbi12i 433 . 2  F : -1-1->  F : -onto->  F :
-->  F `
 F `  F  Fn  ran  F
5 df-3an 886 . . 3  F  Fn  ran  F  F `  F `  F  Fn  ran  F  F `  F `
6 eqimss 2991 . . . . . . 7  ran 
F  ran 
F  C_
76anim2i 324 . . . . . 6  F  Fn  ran  F  F  Fn  ran  F  C_
8 df-f 4849 . . . . . 6  F : -->  F  Fn  ran  F 
C_
97, 8sylibr 137 . . . . 5  F  Fn  ran  F  F : -->
109pm4.71ri 372 . . . 4  F  Fn  ran  F  F : -->  F  Fn  ran  F
1110anbi1i 431 . . 3  F  Fn  ran  F  F `
 F `  F : -->  F  Fn  ran  F  F `
 F `
12 an32 496 . . 3  F : -->  F  Fn  ran  F  F `  F `  F : -->  F `  F `  F  Fn  ran  F
135, 11, 123bitrri 196 . 2  F : -->  F `
 F `  F  Fn  ran  F  F  Fn  ran  F  F `
 F `
141, 4, 133bitri 195 1  F : -1-1-onto->  F  Fn  ran  F  F `
 F `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242  wral 2300    C_ wss 2911   ran crn 4289    Fn wfn 4840   -->wf 4841   -1-1->wf1 4842   -onto->wfo 4843   -1-1-onto->wf1o 4844   ` cfv 4845
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator