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Theorem fun2cnv 4906
Description: The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that A is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
fun2cnv (Fun Ax∃*y xAy)
Distinct variable group:   x,y,A

Proof of Theorem fun2cnv
StepHypRef Expression
1 funcnv2 4902 . 2 (Fun Ax∃*y yAx)
2 vex 2554 . . . . 5 y V
3 vex 2554 . . . . 5 x V
42, 3brcnv 4461 . . . 4 (yAxxAy)
54mobii 1934 . . 3 (∃*y yAx∃*y xAy)
65albii 1356 . 2 (x∃*y yAxx∃*y xAy)
71, 6bitri 173 1 (Fun Ax∃*y xAy)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240  ∃*wmo 1898   class class class wbr 3755  ccnv 4287  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-fun 4847
This theorem is referenced by:  svrelfun  4907  fun11  4909
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