Theorem funcnv 5381

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 5380 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 2802	. . . . . . 7 |- x e. _V
2		vex 2802	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4956	. . . . . 6 |- ( x A y -> y e. ran A )
4	3	pm4.71ri 392	. . . . 5 |- ( x A y <-> ( y e. ran A /\ x A y ) )
5	4	mobii 2114	. . . 4 |- ( E* x x A y <-> E* x ( y e. ran A /\ x A y ) )
6		moanimv 2153	. . . 4 |- ( E* x ( y e. ran A /\ x A y ) <-> ( y e. ran A -> E* x x A y ) )
7	5, 6	bitri 184	. . 3 |- ( E* x x A y <-> ( y e. ran A -> E* x x A y ) )
8	7	albii 1516	. 2 |- ( A. y E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
9		funcnv2 5380	. 2 |- ( Fun `' A <-> A. y E* x x A y )
10		df-ral 2513	. 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 212	1 |- ( Fun `' A <-> A. y e. ran A E* x x A y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   E*wmo 2078   e. wcel 2200   A.wral 2508   class class class wbr 4082   `'ccnv 4717   ran crn 4719   Fun wfun 5311

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319

This theorem is referenced by:  funcnv3  5382  fncnv  5386