Theorem dffun6 3539

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, dffun7 3540 shows that it doesn't matter which meaning we pick.)

Assertion

Ref	Expression
dffun6	|- (Fun A <-> (Rel A /\ A.x e. dom AE*y xAy))

Distinct variable group:   x,y,A

Proof of Theorem dffun6

Step	Hyp	Ref	Expression
1		dffunmo 3531	. 2 |- (Fun A <-> (Rel A /\ A.xE*y xAy))
2		moabs 1415	. . . . . 6 |- (E*y xAy <-> (E.y xAy -> E*y xAy))
3		visset 1813	. . . . . . . 8 |- x e. V
4	3	eldm 3307	. . . . . . 7 |- (x e. dom A <-> E.y xAy)
5	4	imbi1i 186	. . . . . 6 |- ((x e. dom A -> E*y xAy) <-> (E.y xAy -> E*y xAy))
6	2, 5	bitr4 176	. . . . 5 |- (E*y xAy <-> (x e. dom A -> E*y xAy))
7	6	albii 999	. . . 4 |- (A.xE*y xAy <-> A.x(x e. dom A -> E*y xAy))
8		df-ral 1649	. . . 4 |- (A.x e. dom AE*y xAy <-> A.x(x e. dom A -> E*y xAy))
9	7, 8	bitr4 176	. . 3 |- (A.xE*y xAy <-> A.x e. dom AE*y xAy)
10	9	anbi2i 480	. 2 |- ((Rel A /\ A.xE*y xAy) <-> (Rel A /\ A.x e. dom AE*y xAy))
11	1, 10	bitr 173	1 |- (Fun A <-> (Rel A /\ A.x e. dom AE*y xAy))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980  E*wmo 1381  A.wral 1645   class class class wbr 2619  dom cdm 3170  Rel wrel 3175  Fun wfun 3176

This theorem is referenced by:  dffun7 3540  dffun8 3541  brdom5 4802

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-cnv 3186  df-co 3187  df-dm 3188  df-fun 3192

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