Theorem funmo 5605

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 5604	. . . . . 6 |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) )
2	1	simplbi 467	. . . . 5 |- ( Fun F -> Rel F )
3		brrelex 4878	. . . . . 6 |- ( ( Rel F /\ A F y ) -> A e. _V )
4	3	ex 441	. . . . 5 |- ( Rel F -> ( A F y -> A e. _V ) )
5	2, 4	syl 17	. . . 4 |- ( Fun F -> ( A F y -> A e. _V ) )
6	5	ancrd 563	. . 3 |- ( Fun F -> ( A F y -> ( A e. _V /\ A F y ) ) )
7	6	alrimiv 1781	. 2 |- ( Fun F -> A. y ( A F y -> ( A e. _V /\ A F y ) ) )
8		breq1 4398	. . . . . . 7 |- ( x = A -> ( x F y <-> A F y ) )
9	8	mobidv 2340	. . . . . 6 |- ( x = A -> ( E* y x F y <-> E* y A F y ) )
10	9	imbi2d 323	. . . . 5 |- ( x = A -> ( ( Fun F -> E* y x F y ) <-> ( Fun F -> E* y A F y ) ) )
11	1	simprbi 471	. . . . . 6 |- ( Fun F -> A. x E* y x F y )
12	11	19.21bi 1967	. . . . 5 |- ( Fun F -> E* y x F y )
13	10, 12	vtoclg 3093	. . . 4 |- ( A e. _V -> ( Fun F -> E* y A F y ) )
14	13	com12 31	. . 3 |- ( Fun F -> ( A e. _V -> E* y A F y ) )
15		moanimv 2380	. . 3 |- ( E* y ( A e. _V /\ A F y ) <-> ( A e. _V -> E* y A F y ) )
16	14, 15	sylibr 217	. 2 |- ( Fun F -> E* y ( A e. _V /\ A F y ) )
17		moim 2368	. 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 61	1 |- ( Fun F -> E* y A F y )

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   = wceq 1452   e. wcel 1904   E*wmo 2320   _Vcvv 3031   class class class wbr 4395   Rel wrel 4844   Fun wfun 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-fun 5591

This theorem is referenced by:  funeu  5613  funco  5627  fununmo  5632  imadif  5668  fneu  5690  dff3  6050  shftfn  13213