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Theorem funcnv 5653
 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 5652 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 3034 . . . . . . 7
2 vex 3034 . . . . . . 7
31, 2brelrn 5071 . . . . . 6
43pm4.71ri 645 . . . . 5
54mobii 2342 . . . 4
6 moanimv 2380 . . . 4
75, 6bitri 257 . . 3
87albii 1699 . 2
9 funcnv2 5652 . 2
10 df-ral 2761 . 2
118, 9, 103bitr4i 285 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376  wal 1450   wcel 1904  wmo 2320  wral 2756   class class class wbr 4395  ccnv 4838   crn 4840   wfun 5583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-fun 5591 This theorem is referenced by:  funcnv3  5654  fncnv  5657
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