Theorem funcnv 3553

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 3552 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 1810	. . . . . . 7 |- x e. V
2		visset 1810	. . . . . . 7 |- y e. V
3	1, 2	brelrn 3340	. . . . . 6 |- (xAy -> y e. ran A)
4	3	pm4.71ri 637	. . . . 5 |- (xAy <-> (y e. ran A /\ xAy))
5	4	mobii 1404	. . . 4 |- (E*x xAy <-> E*x(y e. ran A /\ xAy))
6		moanimv 1428	. . . 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 998	. 2 |- (A.yE*x xAy <-> A.y(y e. ran A -> E*x xAy))
9		funcnv2 3552	. 2 |- (Fun `'A <-> A.yE*x xAy)
10		df-ral 1647	. 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 953   e. wcel 957  E*wmo 1380  A.wral 1643   class class class wbr 2615  `'ccnv 3165  ran crn 3167  Fun wfun 3172

This theorem is referenced by:  funcnv3 3554  fncnv 3557  bra11 9997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188

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