Theorem fun2cnv 3559

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.

Assertion

Ref	Expression
fun2cnv	|- (Fun `'`'A <-> A.xE*y xAy)

Distinct variable group:   x,y,A

Proof of Theorem fun2cnv

Step	Hyp	Ref	Expression
1		funcnv2 3556	. 2 |- (Fun `'`'A <-> A.xE*y y`'Ax)
2		visset 1813	. . . . 5 |- y e. V
3		visset 1813	. . . . 5 |- x e. V
4	2, 3	brcnv 3299	. . . 4 |- (y`'Ax <-> xAy)
5	4	mobii 1405	. . 3 |- (E*y y`'Ax <-> E*y xAy)
6	5	albii 999	. 2 |- (A.xE*y y`'Ax <-> A.xE*y xAy)
7	1, 6	bitr 173	1 |- (Fun `'`'A <-> A.xE*y xAy)

Syntax hints:   <-> wb 146  A.wal 954  E*wmo 1381   class class class wbr 2619  `'ccnv 3169  Fun wfun 3176

This theorem is referenced by:  svrelfun 3560  fun11 3562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-fun 3192

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