Theorem funcnv 3557

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 xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 3556 for a simpler version.

Assertion

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

Distinct variable group:   x,y,A

Proof of Theorem funcnv

Step	Hyp	Ref	Expression
1		visset 1813	. . . . . . 7 |- x e. V
2		visset 1813	. . . . . . 7 |- y e. V
3	1, 2	brelrn 3344	. . . . . 6 |- (xAy -> y e. ran A)
4	3	pm4.71ri 638	. . . . 5 |- (xAy <-> (y e. ran A /\ xAy))
5	4	mobii 1405	. . . 4 |- (E*x xAy <-> E*x(y e. ran A /\ xAy))
6		moanimv 1429	. . . 4 |- (E*x(y e. ran A /\ xAy) <-> (y e. ran A -> E*x xAy))
7	5, 6	bitr 173	. . 3 |- (E*x xAy <-> (y e. ran A -> E*x xAy))
8	7	albii 999	. 2 |- (A.yE*x xAy <-> A.y(y e. ran A -> E*x xAy))
9		funcnv2 3556	. 2 |- (Fun `'A <-> A.yE*x xAy)
10		df-ral 1649	. 2 |- (A.y e. ran AE*x xAy <-> A.y(y e. ran A -> E*x xAy))
11	8, 9, 10	3bitr4 183	1 |- (Fun `'A <-> A.y e. ran AE*x xAy)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   e. wcel 958  E*wmo 1381  A.wral 1645   class class class wbr 2619  `'ccnv 3169  ran crn 3171  Fun wfun 3176

This theorem is referenced by:  funcnv3 3558  fncnv 3561  bra11 10041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192

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