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Theorem funcnv 5167
 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 5166 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 2730 . . . . . . 7
2 vex 2730 . . . . . . 7
31, 2brelrn 4816 . . . . . 6
43pm4.71ri 617 . . . . 5
54mobii 2149 . . . 4
6 moanimv 2171 . . . 4
75, 6bitri 242 . . 3
87albii 1554 . 2
9 funcnv2 5166 . 2
10 df-ral 2513 . 2
118, 9, 103bitr4i 270 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360  wal 1532   wcel 1621  wmo 2115  wral 2509   class class class wbr 3920  ccnv 4579   crn 4581   wfun 4586 This theorem is referenced by:  funcnv3  5168  fncnv  5171 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-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-fun 4602
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