Theorem dffun8 4448

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 4447. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
dffun8	|- (Fun A <-> (Rel A /\ A.x e. dom AE!y xAy))

Distinct variable group:   x,y,A

Proof of Theorem dffun8

Step	Hyp	Ref	Expression
1		dffun7 4447	. 2 |- (Fun A <-> (Rel A /\ A.x e. dom AE*y xAy))
2		visset 2295	. . . . . . 7 |- x e. _V
3	2	eldm 4153	. . . . . 6 |- (x e. dom A <-> E.y xAy)
4		pm5.5 804	. . . . . 6 |- (E.y xAy -> ((E.y xAy -> E!y xAy) <-> E!y xAy))
5	3, 4	sylbi 216	. . . . 5 |- (x e. dom A -> ((E.y xAy -> E!y xAy) <-> E!y xAy))
6		df-mo 1776	. . . . 5 |- (E*y xAy <-> (E.y xAy -> E!y xAy))
7	5, 6	syl5bb 591	. . . 4 |- (x e. dom A -> (E*y xAy <-> E!y xAy))
8	7	ralbiia 2133	. . 3 |- (A.x e. dom AE*y xAy <-> A.x e. dom AE!y xAy)
9	8	anbi2i 538	. 2 |- ((Rel A /\ A.x e. dom AE*y xAy) <-> (Rel A /\ A.x e. dom AE!y xAy))
10	1, 9	bitri 190	1 |- (Fun A <-> (Rel A /\ A.x e. dom AE!y xAy))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  E.wex 1326  E!weu 1771  E*wmo 1772  A.wral 2105   class class class wbr 3338  dom cdm 3986  Rel wrel 3991  Fun wfun 3992

This theorem is referenced by:  nfunsnOLD 4707

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-cnv 4002  df-co 4003  df-dm 4004  df-fun 4008

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