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Theorem f1cnvcnv 5640
 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 5452 . 2
2 dffn2 5585 . . . 4
3 dmcnvcnv 5085 . . . . 5
4 df-fn 5450 . . . . 5
53, 4mpbiran2 886 . . . 4
62, 5bitr3i 243 . . 3
7 relcnv 5235 . . . . 5
8 dfrel2 5314 . . . . 5
97, 8mpbi 200 . . . 4
109funeqi 5467 . . 3
116, 10anbi12ci 680 . 2
121, 11bitri 241 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652  cvv 2949  ccnv 4870   cdm 4871   wrel 4876   wfun 5441   wfn 5442  wf 5443  wf1 5444 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452
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