Theorem dffun7 4928

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 4929 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 4916	. 2 |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
2		moabs 1949	. . . . . 6 |- ( E* y x A y <-> ( E. y x A y -> E* y x A y ) )
3		vex 2560	. . . . . . . 8 |- x e. _V
4	3	eldm 4532	. . . . . . 7 |- ( x e. dom A <-> E. y x A y )
5	4	imbi1i 227	. . . . . 6 |- ( ( x e. dom A -> E* y x A y ) <-> ( E. y x A y -> E* y x A y ) )
6	2, 5	bitr4i 176	. . . . 5 |- ( E* y x A y <-> ( x e. dom A -> E* y x A y ) )
7	6	albii 1359	. . . 4 |- ( A. x E* y x A y <-> A. x ( x e. dom A -> E* y x A y ) )
8		df-ral 2311	. . . 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 176	. . 3 |- ( A. x E* y x A y <-> A. x e. dom A E* y x A y )
10	9	anbi2i 430	. 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 173	1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   e. wcel 1393   E*wmo 1901   A.wral 2306   class class class wbr 3764   dom cdm 4345   Rel wrel 4350   Fun wfun 4896

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904

This theorem is referenced by:  dffun8  4929  dffun9  4930  funco  4940  funimaexglem  4982  frecuzrdgfn  9198