Theorem funcnv 5958

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 5957 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 3203	. . . . . . 7 |- x e. _V
2		vex 3203	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 5356	. . . . . 6 |- ( x A y -> y e. ran A )
4	3	pm4.71ri 665	. . . . 5 |- ( x A y <-> ( y e. ran A /\ x A y ) )
5	4	mobii 2493	. . . 4 |- ( E* x x A y <-> E* x ( y e. ran A /\ x A y ) )
6		moanimv 2531	. . . 4 |- ( E* x ( y e. ran A /\ x A y ) <-> ( y e. ran A -> E* x x A y ) )
7	5, 6	bitri 264	. . 3 |- ( E* x x A y <-> ( y e. ran A -> E* x x A y ) )
8	7	albii 1747	. 2 |- ( A. y E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
9		funcnv2 5957	. 2 |- ( Fun `' A <-> A. y E* x x A y )
10		df-ral 2917	. 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 292	1 |- ( Fun `' A <-> A. y e. ran A E* x x A y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   A.wral 2912   class class class wbr 4653   `'ccnv 5113   ran crn 5115   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890

This theorem is referenced by:  funcnv3  5959  fncnv  5962