Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1cnvcnv Unicode version

Theorem f1cnvcnv 5794
 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

Proof of Theorem f1cnvcnv
StepHypRef Expression
1 df-f1 5598 . 2
2 dffn2 5737 . . . 4
3 dmcnvcnv 5230 . . . . 5
4 df-fn 5596 . . . . 5
53, 4mpbiran2 919 . . . 4
62, 5bitr3i 251 . . 3
7 relcnv 5379 . . . . 5
8 dfrel2 5462 . . . . 5
97, 8mpbi 208 . . . 4
109funeqi 5613 . . 3
116, 10anbi12ci 698 . 2
121, 11bitri 249 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  /\wa 369  =wceq 1395   cvv 3109  'ccnv 5003  domcdm 5004  Relwrel 5009  Funwfun 5587  Fnwfn 5588  -->wf 5589  -1-1->`wf1 5590 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598
 Copyright terms: Public domain W3C validator