Theorem dffun7 4871

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 4872 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 AE*y xAy))

Distinct variable group:   x,y,A

Proof of Theorem dffun7

Step	Hyp	Ref	Expression
1		dffun6 4859	. 2 |- ( Fun A <-> ( Rel A /\ A.xE*y xAy))
2		moabs 1946	. . . . . 6 |- (E*y xAy <-> (E.y xAy -> E*y xAy))
3		vex 2554	. . . . . . . 8 |- x e. _V
4	3	eldm 4475	. . . . . . 7 |- (x e. dom A <-> E.y xAy)
5	4	imbi1i 227	. . . . . 6 |- ((x e. dom A -> E*y xAy) <-> (E.y xAy -> E*y xAy))
6	2, 5	bitr4i 176	. . . . 5 |- (E*y xAy <-> (x e. dom A -> E*y xAy))
7	6	albii 1356	. . . 4 |- (A.xE*y xAy <-> A.x(x e. dom A -> E*y xAy))
8		df-ral 2305	. . . 4 |- (A.x e. dom AE*y xAy <-> A.x(x e. dom A -> E*y xAy))
9	7, 8	bitr4i 176	. . 3 |- (A.xE*y xAy <-> A.x e. dom AE*y xAy)
10	9	anbi2i 430	. 2 |- (( Rel A /\ A.xE*y xAy) <-> ( Rel A /\ A.x e. dom AE*y xAy))
11	1, 10	bitri 173	1 |- ( Fun A <-> ( Rel A /\ A.x e. dom AE*y xAy))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378   e. wcel 1390  E*wmo 1898  A.wral 2300   class class class wbr 3755   dom cdm 4288   Rel wrel 4293   Fun wfun 4839

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847

This theorem is referenced by:  dffun8  4872  dffun9  4873  funco  4883  funimaexglem  4925  frecuzrdgfn  8879