Theorem fun2cnv 5440

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 5436	. 2 |- ( Fun `' `' A <-> A. x E* y y `' A x )
2		vex 2824	. . . . 5 |- y e. _V
3		vex 2824	. . . . 5 |- x e. _V
4	2, 3	brcnv 4958	. . . 4 |- ( y `' A x <-> x A y )
5	4	mobii 2123	. . 3 |- ( E* y y `' A x <-> E* y x A y )
6	5	albii 1523	. 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 1400   E*wmo 2087   class class class wbr 4125   `'ccnv 4768   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-fun 5374

This theorem is referenced by:  svrelfun  5441  fun11  5443