Theorem dffun7 5299

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 5300 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 5286	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		moabs 2103	. . . . . 6 |- ( E* y x A y <-> ( E. y x A y -> E* y x A y ) )
3		vex 2775	. . . . . . . 8 |- x e. _V
4	3	eldm 4876	. . . . . . 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 1493	. . . 4 |- ( A. x E* y x A y <-> A. x ( x e. dom A -> E* y x A y ) )
8		df-ral 2489	. . . 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 1371   E.wex 1515   E*wmo 2055   e. wcel 2176   A.wral 2484   class class class wbr 4045   dom cdm 4676   Rel wrel 4681   Fun wfun 5266

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-id 4341  df-cnv 4684  df-co 4685  df-dm 4686  df-fun 5274

This theorem is referenced by:  dffun8  5300  dffun9  5301  funco  5312  funimaexglem  5358  frecuzrdgtcl  10559  frecuzrdgfunlem  10566  imasaddfnlemg  13179