Theorem funeu 3523

Description: There is exactly one value of a function.

Assertion

Ref	Expression
funeu	|- ((Fun F /\ xFy) -> E!y xFy)

Distinct variable group:   x,y,F

Proof of Theorem funeu

Step	Hyp	Ref	Expression
1		19.8a 1025	. . . 4 |- (xFy -> E.y xFy)
2		dffun3 3513	. . . . . 6 |- (Fun F <-> (Rel F /\ A.xE.zA.y(xFy -> y = z)))
3	2	pm3.27bi 326	. . . . 5 |- (Fun F -> A.xE.zA.y(xFy -> y = z))
4	3	19.21bi 1056	. . . 4 |- (Fun F -> E.zA.y(xFy -> y = z))
5	1, 4	anim12i 333	. . 3 |- ((xFy /\ Fun F) -> (E.y xFy /\ E.zA.y(xFy -> y = z)))
6		ax-17 968	. . . 4 |- (xFy -> A.z xFy)
7	6	eu3 1390	. . 3 |- (E!y xFy <-> (E.y xFy /\ E.zA.y(xFy -> y = z)))
8	5, 7	sylibr 200	. 2 |- ((xFy /\ Fun F) -> E!y xFy)
9	8	ancoms 436	1 |- ((Fun F /\ xFy) -> E!y xFy)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  E!weu 1373   class class class wbr 2609  Rel wrel 3165  Fun wfun 3166

This theorem is referenced by:  funeu2 3524  fneu 3578  funbrfv 3735

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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