Theorem funcnv 5231

Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that x A y. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5230 for a simpler version. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
funcnv	|- ( Fun `' A <-> A. y e. ran A E* x x A y )

Distinct variable group:   x, y, A

Proof of Theorem funcnv

Step	Hyp	Ref	Expression
1		vex 2715	. . . . . . 7 |- x e. _V
2		vex 2715	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4819	. . . . . 6 |- ( x A y -> y e. ran A )
4	3	pm4.71ri 390	. . . . 5 |- ( x A y <-> ( y e. ran A /\ x A y ) )
5	4	mobii 2043	. . . 4 |- ( E* x x A y <-> E* x ( y e. ran A /\ x A y ) )
6		moanimv 2081	. . . 4 |- ( E* x ( y e. ran A /\ x A y ) <-> ( y e. ran A -> E* x x A y ) )
7	5, 6	bitri 183	. . 3 |- ( E* x x A y <-> ( y e. ran A -> E* x x A y ) )
8	7	albii 1450	. 2 |- ( A. y E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
9		funcnv2 5230	. 2 |- ( Fun `' A <-> A. y E* x x A y )
10		df-ral 2440	. 2 |- ( A. y e. ran A E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
11	8, 9, 10	3bitr4i 211	1 |- ( Fun `' A <-> A. y e. ran A E* x x A y )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   E*wmo 2007   e. wcel 2128   A.wral 2435   class class class wbr 3965   `'ccnv 4585   ran crn 4587   Fun wfun 5164

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-fun 5172

This theorem is referenced by:  funcnv3  5232  fncnv  5236