Theorem dffun7 5028

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 5029 shows that it doesn't 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 5016	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		moabs 1997	. . . . . 6 |- ( E* y x A y <-> ( E. y x A y -> E* y x A y ) )
3		vex 2622	. . . . . . . 8 |- x e. _V
4	3	eldm 4621	. . . . . . 7 |- ( x e. dom A <-> E. y x A y )
5	4	imbi1i 236	. . . . . 6 |- ( ( x e. dom A -> E* y x A y ) <-> ( E. y x A y -> E* y x A y ) )
6	2, 5	bitr4i 185	. . . . 5 |- ( E* y x A y <-> ( x e. dom A -> E* y x A y ) )
7	6	albii 1404	. . . 4 |- ( A. x E* y x A y <-> A. x ( x e. dom A -> E* y x A y ) )
8		df-ral 2364	. . . 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 185	. . 3 |- ( A. x E* y x A y <-> A. x e. dom A E* y x A y )
10	9	anbi2i 445	. 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 182	1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1287   E.wex 1426   e. wcel 1438   E*wmo 1949   A.wral 2359   class class class wbr 3837   dom cdm 4428   Rel wrel 4433   Fun wfun 4996

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004

This theorem is referenced by:  dffun8  5029  dffun9  5030  funco  5040  funimaexglem  5083  frecuzrdgtcl  9784  frecuzrdgfunlem  9791