Theorem fun2cnv 5251

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 5247	. 2 |- ( Fun `' `' A <-> A. x E* y y `' A x )
2		vex 2728	. . . . 5 |- y e. _V
3		vex 2728	. . . . 5 |- x e. _V
4	2, 3	brcnv 4786	. . . 4 |- ( y `' A x <-> x A y )
5	4	mobii 2051	. . 3 |- ( E* y y `' A x <-> E* y x A y )
6	5	albii 1458	. 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 1341   E*wmo 2015   class class class wbr 3981   `'ccnv 4602   Fun wfun 5181

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982  df-opab 4043  df-id 4270  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-fun 5189

This theorem is referenced by:  svrelfun  5252  fun11  5254