Theorem fun2cnv 4983

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 `' `' A <-> A. x E* y x A y )

Distinct variable group:   x, y, A

Proof of Theorem fun2cnv

Step	Hyp	Ref	Expression
1		funcnv2 4979	. 2 |- ( Fun `' `' A <-> A. x E* y y `' A x )
2		vex 2604	. . . . 5 |- y e. _V
3		vex 2604	. . . . 5 |- x e. _V
4	2, 3	brcnv 4536	. . . 4 |- ( y `' A x <-> x A y )
5	4	mobii 1978	. . 3 |- ( E* y y `' A x <-> E* y x A y )
6	5	albii 1399	. 2 |- ( A. x E* y y `' A x <-> A. x E* y x A y )
7	1, 6	bitri 182	1 |- ( Fun `' `' A <-> A. x E* y x A y )

Syntax hints:   <-> wb 103   A.wal 1282   E*wmo 1942   class class class wbr 3785   `'ccnv 4362   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-fun 4924

This theorem is referenced by:  svrelfun  4984  fun11  4986