Theorem dffn5im 5460

Description: Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 5156 and dmmptss 5030. (Contributed by Jim Kingdon, 31-Dec-2018.)

Assertion

Ref	Expression
dffn5im	|- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Distinct variable groups:   x, A   x, F

Proof of Theorem dffn5im

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrel 5216	. . . 4 |- ( F Fn A -> Rel F )
2		dfrel4v 4985	. . . 4 |- ( Rel F <-> F = { <. x , y >. | x F y } )
3	1, 2	sylib 121	. . 3 |- ( F Fn A -> F = { <. x , y >. | x F y } )
4		fnbr 5220	. . . . . . 7 |- ( ( F Fn A /\ x F y ) -> x e. A )
5	4	ex 114	. . . . . 6 |- ( F Fn A -> ( x F y -> x e. A ) )
6	5	pm4.71rd 391	. . . . 5 |- ( F Fn A -> ( x F y <-> ( x e. A /\ x F y ) ) )
7		eqcom 2139	. . . . . . 7 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
8		fnbrfvb 5455	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
9	7, 8	syl5bb 191	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
10	9	pm5.32da 447	. . . . 5 |- ( F Fn A -> ( ( x e. A /\ y = ( F ` x ) ) <-> ( x e. A /\ x F y ) ) )
11	6, 10	bitr4d 190	. . . 4 |- ( F Fn A -> ( x F y <-> ( x e. A /\ y = ( F ` x ) ) ) )
12	11	opabbidv 3989	. . 3 |- ( F Fn A -> { <. x , y >. | x F y } = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
13	3, 12	eqtrd 2170	. 2 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
14		df-mpt 3986	. 2 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
15	13, 14	syl6eqr 2188	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   class class class wbr 3924   {copab 3983   |-> cmpt 3984   Rel wrel 4539   Fn wfn 5113   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  fnrnfv  5461  feqmptd  5467  dffn5imf  5469  eqfnfv  5511  fndmin  5520  fcompt  5583  resfunexg  5634  eufnfv  5641  fnovim  5872  offveqb  5994  caofinvl  5997  oprabco  6107  df1st2  6109  df2nd2  6110  xpen  6732  cnmpt1st  12446  cnmpt2nd  12447