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Theorem f1cnvcnv 5339
 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 5128 . 2
2 dffn2 5274 . . . 4
3 dmcnvcnv 4763 . . . . 5
4 df-fn 5126 . . . . 5
53, 4mpbiran2 925 . . . 4
62, 5bitr3i 185 . . 3
7 relcnv 4917 . . . . 5
8 dfrel2 4989 . . . . 5
97, 8mpbi 144 . . . 4
109funeqi 5144 . . 3
116, 10anbi12ci 456 . 2
121, 11bitri 183 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104   wceq 1331  cvv 2686  ccnv 4538   cdm 4539   wrel 4544   wfun 5117   wfn 5118  wf 5119  wf1 5120 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128 This theorem is referenced by: (None)
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