Theorem dffun7 5262

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 5263 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Assertion

Ref	Expression
dffun7	|- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )

Distinct variable group:   x, y, A

Proof of Theorem dffun7

Step	Hyp	Ref	Expression
1		dffun6 5249	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		moabs 2087	. . . . . 6 |- ( E* y x A y <-> ( E. y x A y -> E* y x A y ) )
3		vex 2755	. . . . . . . 8 |- x e. _V
4	3	eldm 4842	. . . . . . 7 |- ( x e. dom A <-> E. y x A y )
5	4	imbi1i 238	. . . . . 6 |- ( ( x e. dom A -> E* y x A y ) <-> ( E. y x A y -> E* y x A y ) )
6	2, 5	bitr4i 187	. . . . 5 |- ( E* y x A y <-> ( x e. dom A -> E* y x A y ) )
7	6	albii 1481	. . . 4 |- ( A. x E* y x A y <-> A. x ( x e. dom A -> E* y x A y ) )
8		df-ral 2473	. . . 4 |- ( A. x e. dom A E* y x A y <-> A. x ( x e. dom A -> E* y x A y ) )
9	7, 8	bitr4i 187	. . 3 |- ( A. x E* y x A y <-> A. x e. dom A E* y x A y )
10	9	anbi2i 457	. 2 |- ( ( Rel A /\ A. x E* y x A y ) <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
11	1, 10	bitri 184	1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1503   E*wmo 2039   e. wcel 2160   A.wral 2468   class class class wbr 4018   dom cdm 4644   Rel wrel 4649   Fun wfun 5229

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-cnv 4652  df-co 4653  df-dm 4654  df-fun 5237

This theorem is referenced by:  dffun8  5263  dffun9  5264  funco  5275  funimaexglem  5318  frecuzrdgtcl  10443  frecuzrdgfunlem  10450  imasaddfnlemg  12791