Theorem funeu 3543

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 1031	. . . 4 |- (xFy -> E.y xFy)
2		dffun3 3533	. . . . . 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 1062	. . . 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-17 973	. . . 4 |- (xFy -> A.z xFy)
7	6	eu3 1399	. . 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 438	1 |- ((Fun F /\ xFy) -> E!y xFy)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958  E.wex 982  E!weu 1382   class class class wbr 2624  Rel wrel 3181  Fun wfun 3182

This theorem is referenced by:  funeu2 3544  fneu 3598  funbrfv 3756

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-cnv 3192  df-co 3193  df-fun 3198

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