Theorem dffun3 3513

Description: Alternate definition of function.

Assertion

Ref	Expression
dffun3	|- (Fun A <-> (Rel A /\ A.xE.zA.y(xAy -> y = z)))

Distinct variable group:   x,y,z,A

Proof of Theorem dffun3

Step	Hyp	Ref	Expression
1		dffun2 3512	. 2 |- (Fun A <-> (Rel A /\ A.xA.yA.z((xAy /\ xAz) -> y = z)))
2		breq2 2613	. . . . . 6 |- (y = z -> (xAy <-> xAz))
3	2	mo4 1396	. . . . 5 |- (E*y xAy <-> A.yA.z((xAy /\ xAz) -> y = z))
4		ax-17 968	. . . . . 6 |- (xAy -> A.z xAy)
5	4	mo2 1393	. . . . 5 |- (E*y xAy <-> E.zA.y(xAy -> y = z))
6	3, 5	bitr3 175	. . . 4 |- (A.yA.z((xAy /\ xAz) -> y = z) <-> E.zA.y(xAy -> y = z))
7	6	albii 996	. . 3 |- (A.xA.yA.z((xAy /\ xAz) -> y = z) <-> A.xE.zA.y(xAy -> y = z))
8	7	anbi2i 479	. 2 |- ((Rel A /\ A.xA.yA.z((xAy /\ xAz) -> y = z)) <-> (Rel A /\ A.xE.zA.y(xAy -> y = z)))
9	1, 8	bitr 173	1 |- (Fun A <-> (Rel A /\ A.xE.zA.y(xAy -> y = z)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  E*wmo 1374   class class class wbr 2609  Rel wrel 3165  Fun wfun 3166

This theorem is referenced by:  dffun5 3515  dffunmof 3516  funeu 3523

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-cnv 3176  df-co 3177  df-fun 3182

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