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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 3112 . . . . . . 7
2 vex 3112 . . . . . . 7
31, 2brelrn 5238 . . . . . 6
43pm4.71ri 633 . . . . 5
54mobii 2307 . . . 4
6 moanimv 2352 . . . 4
75, 6bitri 249 . . 3
87albii 1640 . 2
9 funcnv2 5652 . 2
10 df-ral 2812 . 2
118, 9, 103bitr4i 277 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  e.wcel 1818  E*wmo 2283  A.wral 2807   class class class wbr 4452  `'ccnv 5003  rancrn 5005  Funwfun 5587
This theorem is referenced by:  funcnv3  5654  fncnv  5657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-fun 5595
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