Theorem funmo 4860

Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Assertion

Ref	Expression
funmo	|- ( Fun F -> E*y A Fy)

Distinct variable groups:   y,A   y, F

Proof of Theorem funmo

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun6 4859	. . . . . 6 |- ( Fun F <-> ( Rel F /\ A.xE*y x Fy))
2	1	simplbi 259	. . . . 5 |- ( Fun F -> Rel F)
3		brrelex 4325	. . . . . 6 |- (( Rel F /\ A Fy) -> A e. _V)
4	3	ex 108	. . . . 5 |- ( Rel F -> (A Fy -> A e. _V))
5	2, 4	syl 14	. . . 4 |- ( Fun F -> (A Fy -> A e. _V))
6	5	ancrd 309	. . 3 |- ( Fun F -> (A Fy -> (A e. _V /\ A Fy)))
7	6	alrimiv 1751	. 2 |- ( Fun F -> A.y(A Fy -> (A e. _V /\ A Fy)))
8		breq1 3758	. . . . . . 7 |- (x = A -> (x Fy <-> A Fy))
9	8	mobidv 1933	. . . . . 6 |- (x = A -> (E*y x Fy <-> E*y A Fy))
10	9	imbi2d 219	. . . . 5 |- (x = A -> (( Fun F -> E*y x Fy) <-> ( Fun F -> E*y A Fy)))
11	1	simprbi 260	. . . . . 6 |- ( Fun F -> A.xE*y x Fy)
12	11	19.21bi 1447	. . . . 5 |- ( Fun F -> E*y x Fy)
13	10, 12	vtoclg 2607	. . . 4 |- (A e. _V -> ( Fun F -> E*y A Fy))
14	13	com12 27	. . 3 |- ( Fun F -> (A e. _V -> E*y A Fy))
15		moanimv 1972	. . 3 |- (E*y(A e. _V /\ A Fy) <-> (A e. _V -> E*y A Fy))
16	14, 15	sylibr 137	. 2 |- ( Fun F -> E*y(A e. _V /\ A Fy))
17		moim 1961	. 2 |- (A.y(A Fy -> (A e. _V /\ A Fy)) -> (E*y(A e. _V /\ A Fy) -> E*y A Fy))
18	7, 16, 17	sylc 56	1 |- ( Fun F -> E*y A Fy)

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   = wceq 1242   e. wcel 1390  E*wmo 1898   _Vcvv 2551   class class class wbr 3755   Rel wrel 4293   Fun wfun 4839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847

This theorem is referenced by:  funeu  4869  funco  4883  imadif  4922  fneu  4946  dff3im  5255