Theorem funcnv 5192

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 5191 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 2692	. . . . . . 7 |- x e. _V
2		vex 2692	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4780	. . . . . 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 2037	. . . 4 |- ( E* x x A y <-> E* x ( y e. ran A /\ x A y ) )
6		moanimv 2075	. . . 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 1447	. 2 |- ( A. y E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
9		funcnv2 5191	. 2 |- ( Fun `' A <-> A. y E* x x A y )
10		df-ral 2422	. 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 1330   e. wcel 1481   E*wmo 2001   A.wral 2417   class class class wbr 3937   `'ccnv 4546   ran crn 4548   Fun wfun 5125

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133

This theorem is referenced by:  funcnv3  5193  fncnv  5197