Theorem funcnv 5249

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 5248 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 2729	. . . . . . 7 |- x e. _V
2		vex 2729	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4837	. . . . . 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 2051	. . . 4 |- ( E* x x A y <-> E* x ( y e. ran A /\ x A y ) )
6		moanimv 2089	. . . 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 1458	. 2 |- ( A. y E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
9		funcnv2 5248	. 2 |- ( Fun `' A <-> A. y E* x x A y )
10		df-ral 2449	. 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 1341   E*wmo 2015   e. wcel 2136   A.wral 2444   class class class wbr 3982   `'ccnv 4603   ran crn 4605   Fun wfun 5182

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190

This theorem is referenced by:  funcnv3  5250  fncnv  5254