Theorem fun2cnv 5420

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 5416	. 2 |- ( Fun `' `' A <-> A. x E* y y `' A x )
2		vex 2816	. . . . 5 |- y e. _V
3		vex 2816	. . . . 5 |- x e. _V
4	2, 3	brcnv 4938	. . . 4 |- ( y `' A x <-> x A y )
5	4	mobii 2117	. . 3 |- ( E* y y `' A x <-> E* y x A y )
6	5	albii 1519	. 2 |- ( A. x E* y y `' A x <-> A. x E* y x A y )
7	1, 6	bitri 184	1 |- ( Fun `' `' A <-> A. x E* y x A y )

Syntax hints:   <-> wb 105   A.wal 1396   E*wmo 2081   class class class wbr 4109   `'ccnv 4748   Fun wfun 5346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-fun 5354

This theorem is referenced by:  svrelfun  5421  fun11  5423