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

Theorem f1cnvcnv 5302
 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 4605 . 2
2 dffn2 5247 . . . 4
3 dmcnvcnv 4808 . . . . 5
4 df-fn 4603 . . . . 5
53, 4mpbiran2 890 . . . 4
62, 5bitr3i 244 . . 3
7 relcnv 4958 . . . . 5
8 dfrel2 5031 . . . . 5
97, 8mpbi 201 . . . 4
109funeqi 5133 . . 3
116, 10anbi12ci 682 . 2
121, 11bitri 242 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1619  cvv 2727  ccnv 4579   cdm 4580   wrel 4585   wfun 4586   wfn 4587  wf 4588  wf1 4589 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605
 Copyright terms: Public domain W3C validator