Theorem dffun6 3531

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 3532 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 3523	. 2 |- (Fun A <-> (Rel A /\ A.xE*y xAy))
2		moabs 1413	. . . . . 6 |- (E*y xAy <-> (E.y xAy -> E*y xAy))
3		visset 1809	. . . . . . . 8 |- x e. V
4	3	eldm 3302	. . . . . . 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 997	. . . 4 |- (A.xE*y xAy <-> A.x(x e. dom A -> E*y xAy))
8		df-ral 1646	. . . 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 952   e. wcel 956  E.wex 978  E*wmo 1379  A.wral 1642   class class class wbr 2614  dom cdm 3165  Rel wrel 3170  Fun wfun 3171

This theorem is referenced by:  dffun7 3532  dffun8 3533  brdom5 4782

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-cnv 3181  df-co 3182  df-dm 3183  df-fun 3187

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