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Theorem funcnv 4903
Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 4902 for a simpler version. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
funcnv (Fun Ay ran A∃*x xAy)
Distinct variable group:   x,y,A

Proof of Theorem funcnv
StepHypRef Expression
1 vex 2554 . . . . . . 7 x V
2 vex 2554 . . . . . . 7 y V
31, 2brelrn 4510 . . . . . 6 (xAyy ran A)
43pm4.71ri 372 . . . . 5 (xAy ↔ (y ran A xAy))
54mobii 1934 . . . 4 (∃*x xAy∃*x(y ran A xAy))
6 moanimv 1972 . . . 4 (∃*x(y ran A xAy) ↔ (y ran A∃*x xAy))
75, 6bitri 173 . . 3 (∃*x xAy ↔ (y ran A∃*x xAy))
87albii 1356 . 2 (y∃*x xAyy(y ran A∃*x xAy))
9 funcnv2 4902 . 2 (Fun Ay∃*x xAy)
10 df-ral 2305 . 2 (y ran A∃*x xAyy(y ran A∃*x xAy))
118, 9, 103bitr4i 201 1 (Fun Ay ran A∃*x xAy)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  ∃*wmo 1898  wral 2300   class class class wbr 3755  ccnv 4287  ran crn 4289  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847
This theorem is referenced by:  funcnv3  4904  fncnv  4908
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