Theorem funeu 4444

Description: There is exactly one value of a function. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)

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		funmo 4437	. . . 4 |- (Fun F -> E*y xFy)
2		df-mo 1776	. . . 4 |- (E*y xFy <-> (E.y xFy -> E!y xFy))
3	1, 2	sylib 215	. . 3 |- (Fun F -> (E.y xFy -> E!y xFy))
4		19.8a 1376	. . 3 |- (xFy -> E.y xFy)
5	3, 4	syl5 20	. 2 |- (Fun F -> (xFy -> E!y xFy))
6	5	imp 377	1 |- ((Fun F /\ xFy) -> E!y xFy)

Syntax hints:   -> wi 3   /\ wa 240  E.wex 1326  E!weu 1771  E*wmo 1772   class class class wbr 3338  Fun wfun 3992

This theorem is referenced by:  funeu2 4446  fneuOLD 4518  funbrfv 4709

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-cnv 4002  df-co 4003  df-fun 4008

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