Theorem dffun7 5223

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 5224 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 5210	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		moabs 2068	. . . . . 6 |- ( E* y x A y <-> ( E. y x A y -> E* y x A y ) )
3		vex 2733	. . . . . . . 8 |- x e. _V
4	3	eldm 4806	. . . . . . 7 |- ( x e. dom A <-> E. y x A y )
5	4	imbi1i 237	. . . . . 6 |- ( ( x e. dom A -> E* y x A y ) <-> ( E. y x A y -> E* y x A y ) )
6	2, 5	bitr4i 186	. . . . 5 |- ( E* y x A y <-> ( x e. dom A -> E* y x A y ) )
7	6	albii 1463	. . . 4 |- ( A. x E* y x A y <-> A. x ( x e. dom A -> E* y x A y ) )
8		df-ral 2453	. . . 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 186	. . 3 |- ( A. x E* y x A y <-> A. x e. dom A E* y x A y )
10	9	anbi2i 454	. 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 183	1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   E.wex 1485   E*wmo 2020   e. wcel 2141   A.wral 2448   class class class wbr 3987   dom cdm 4609   Rel wrel 4614   Fun wfun 5190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-id 4276  df-cnv 4617  df-co 4618  df-dm 4619  df-fun 5198

This theorem is referenced by:  dffun8  5224  dffun9  5225  funco  5236  funimaexglem  5279  frecuzrdgtcl  10355  frecuzrdgfunlem  10362