Theorem funmo 5367

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 F y )

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 5366	. . . . . 6 |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) )
2	1	simplbi 274	. . . . 5 |- ( Fun F -> Rel F )
3		brrelex 4790	. . . . . 6 |- ( ( Rel F /\ A F y ) -> A e. _V )
4	3	ex 115	. . . . 5 |- ( Rel F -> ( A F y -> A e. _V ) )
5	2, 4	syl 14	. . . 4 |- ( Fun F -> ( A F y -> A e. _V ) )
6	5	ancrd 326	. . 3 |- ( Fun F -> ( A F y -> ( A e. _V /\ A F y ) ) )
7	6	alrimiv 1923	. 2 |- ( Fun F -> A. y ( A F y -> ( A e. _V /\ A F y ) ) )
8		breq1 4112	. . . . . . 7 |- ( x = A -> ( x F y <-> A F y ) )
9	8	mobidv 2116	. . . . . 6 |- ( x = A -> ( E* y x F y <-> E* y A F y ) )
10	9	imbi2d 230	. . . . 5 |- ( x = A -> ( ( Fun F -> E* y x F y ) <-> ( Fun F -> E* y A F y ) ) )
11	1	simprbi 275	. . . . . 6 |- ( Fun F -> A. x E* y x F y )
12	11	19.21bi 1607	. . . . 5 |- ( Fun F -> E* y x F y )
13	10, 12	vtoclg 2875	. . . 4 |- ( A e. _V -> ( Fun F -> E* y A F y ) )
14	13	com12 30	. . 3 |- ( Fun F -> ( A e. _V -> E* y A F y ) )
15		moanimv 2156	. . 3 |- ( E* y ( A e. _V /\ A F y ) <-> ( A e. _V -> E* y A F y ) )
16	14, 15	sylibr 134	. 2 |- ( Fun F -> E* y ( A e. _V /\ A F y ) )
17		moim 2145	. 2 |- ( A. y ( A F y -> ( A e. _V /\ A F y ) ) -> ( E* y ( A e. _V /\ A F y ) -> E* y A F y ) )
18	7, 16, 17	sylc 62	1 |- ( Fun F -> E* y A F y )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   = wceq 1398   E*wmo 2081   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   Rel wrel 4754   Fun wfun 5346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-fun 5354

This theorem is referenced by:  funeu  5377  funco  5392  fununmo  5398  imadif  5436  fneu  5462  dff3im  5822  shftfn  11509