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Theorem f1cnvcnv 5445
 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 5260 . 2
2 dffn2 5390 . . . 4
3 dmcnvcnv 4901 . . . . 5
4 df-fn 5258 . . . . 5
53, 4mpbiran2 885 . . . 4
62, 5bitr3i 242 . . 3
7 relcnv 5051 . . . . 5
8 dfrel2 5124 . . . . 5
97, 8mpbi 199 . . . 4
109funeqi 5275 . . 3
116, 10anbi12ci 679 . 2
121, 11bitri 240 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   wceq 1623  cvv 2788  ccnv 4688   cdm 4689   wrel 4694   wfun 5249   wfn 5250  wf 5251  wf1 5252 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260
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