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Theorem funcnv 5310
 Description: The converse of a class is a function iff the class is single-rooted, which means that for any in the range of there is at most one such that . Definition of single-rooted in [Enderton] p. 43. See funcnv2 5309 for a simpler version. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
funcnv
Distinct variable group:   ,,

Proof of Theorem funcnv
StepHypRef Expression
1 vex 2791 . . . . . . 7
2 vex 2791 . . . . . . 7
31, 2brelrn 4909 . . . . . 6
43pm4.71ri 614 . . . . 5
54mobii 2179 . . . 4
6 moanimv 2201 . . . 4
75, 6bitri 240 . . 3
87albii 1553 . 2
9 funcnv2 5309 . 2
10 df-ral 2548 . 2
118, 9, 103bitr4i 268 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1527   wcel 1684  wmo 2144  wral 2543   class class class wbr 4023  ccnv 4688   crn 4690   wfun 5249 This theorem is referenced by:  funcnv3  5311  fncnv  5314 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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257
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