Theorem fun2cnv 5195

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 5191	. 2 |- ( Fun `' `' A <-> A. x E* y y `' A x )
2		vex 2692	. . . . 5 |- y e. _V
3		vex 2692	. . . . 5 |- x e. _V
4	2, 3	brcnv 4730	. . . 4 |- ( y `' A x <-> x A y )
5	4	mobii 2037	. . 3 |- ( E* y y `' A x <-> E* y x A y )
6	5	albii 1447	. 2 |- ( A. x E* y y `' A x <-> A. x E* y x A y )
7	1, 6	bitri 183	1 |- ( Fun `' `' A <-> A. x E* y x A y )

Syntax hints:   <-> wb 104   A.wal 1330   E*wmo 2001   class class class wbr 3937   `'ccnv 4546   Fun wfun 5125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-fun 5133

This theorem is referenced by:  svrelfun  5196  fun11  5198