ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1cnvcnv Unicode version

Theorem f1cnvcnv 5043
Description: Two ways to express that a set (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1cnvcnv  `' `' : dom  -1-1-> _V  Fun  `'  Fun  `' `'

Proof of Theorem f1cnvcnv
StepHypRef Expression
1 df-f1 4850 . 2  `' `' : dom  -1-1-> _V  `' `' : dom  --> _V  Fun  `' `' `'
2 dffn2 4990 . . . 4  `' `'  Fn  dom  `' `' : dom  --> _V
3 dmcnvcnv 4501 . . . . 5  dom  `' `'  dom
4 df-fn 4848 . . . . 5  `' `'  Fn  dom  Fun  `' `'  dom  `' `'  dom
53, 4mpbiran2 847 . . . 4  `' `'  Fn  dom  Fun  `' `'
62, 5bitr3i 175 . . 3  `' `' : dom  --> _V  Fun  `' `'
7 relcnv 4646 . . . . 5  Rel  `'
8 dfrel2 4714 . . . . 5  Rel  `'  `' `' `'  `'
97, 8mpbi 133 . . . 4  `' `' `'  `'
109funeqi 4865 . . 3  Fun  `' `' `'  Fun  `'
116, 10anbi12ci 434 . 2  `' `' : dom  --> _V  Fun  `' `' `'  Fun  `'  Fun  `' `'
121, 11bitri 173 1  `' `' : dom  -1-1-> _V  Fun  `'  Fun  `' `'
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242   _Vcvv 2551   `'ccnv 4287   dom cdm 4288   Rel wrel 4293   Fun wfun 4839    Fn wfn 4840   -->wf 4841   -1-1->wf1 4842
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-bndl 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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator