Theorem funeu 3537

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 1029	. . . 4 |- (xFy -> E.y xFy)
2		dffun3 3527	. . . . . 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 1060	. . . 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-17 971	. . . 4 |- (xFy -> A.z xFy)
7	6	eu3 1397	. . 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 954   = wceq 956  E.wex 980  E!weu 1380   class class class wbr 2619  Rel wrel 3175  Fun wfun 3176

This theorem is referenced by:  funeu2 3538  fneu 3592  funbrfv 3750

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-cnv 3186  df-co 3187  df-fun 3192

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