Theorem dffun4 3514

Description: Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24.

Assertion

Ref	Expression
dffun4	|- (Fun A <-> (Rel A /\ A.xA.yA.z((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z)))

Distinct variable group:   x,y,z,A

Proof of Theorem dffun4

Step	Hyp	Ref	Expression
1		dffun2 3512	. 2 |- (Fun A <-> (Rel A /\ A.xA.yA.z((xAy /\ xAz) -> y = z)))
2		df-br 2610	. . . . . . 7 |- (xAy <-> <.x, y>. e. A)
3		df-br 2610	. . . . . . 7 |- (xAz <-> <.x, z>. e. A)
4	2, 3	anbi12i 481	. . . . . 6 |- ((xAy /\ xAz) <-> (<.x, y>. e. A /\ <.x, z>. e. A))
5	4	imbi1i 186	. . . . 5 |- (((xAy /\ xAz) -> y = z) <-> ((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z))
6	5	albii 996	. . . 4 |- (A.z((xAy /\ xAz) -> y = z) <-> A.z((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z))
7	6	2albii 997	. . 3 |- (A.xA.yA.z((xAy /\ xAz) -> y = z) <-> A.xA.yA.z((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z))
8	7	anbi2i 479	. 2 |- ((Rel A /\ A.xA.yA.z((xAy /\ xAz) -> y = z)) <-> (Rel A /\ A.xA.yA.z((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z)))
9	1, 8	bitr 173	1 |- (Fun A <-> (Rel A /\ A.xA.yA.z((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  <.cop 2401   class class class wbr 2609  Rel wrel 3165  Fun wfun 3166

This theorem is referenced by:  funsn 3529  funopg 3533  funun 3540  fununi 3549  tfrlem7 3902

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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