Theorem dffn5im 5679

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 5356 and dmmptss 5225. (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 5419	. . . 4 |- ( F Fn A -> Rel F )
2		dfrel4v 5180	. . . 4 |- ( Rel F <-> F = { <. x , y >. | x F y } )
3	1, 2	sylib 122	. . 3 |- ( F Fn A -> F = { <. x , y >. | x F y } )
4		fnbr 5425	. . . . . . 7 |- ( ( F Fn A /\ x F y ) -> x e. A )
5	4	ex 115	. . . . . 6 |- ( F Fn A -> ( x F y -> x e. A ) )
6	5	pm4.71rd 394	. . . . 5 |- ( F Fn A -> ( x F y <-> ( x e. A /\ x F y ) ) )
7		eqcom 2231	. . . . . . 7 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
8		fnbrfvb 5672	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
9	7, 8	bitrid 192	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
10	9	pm5.32da 452	. . . . 5 |- ( F Fn A -> ( ( x e. A /\ y = ( F ` x ) ) <-> ( x e. A /\ x F y ) ) )
11	6, 10	bitr4d 191	. . . 4 |- ( F Fn A -> ( x F y <-> ( x e. A /\ y = ( F ` x ) ) ) )
12	11	opabbidv 4150	. . 3 |- ( F Fn A -> { <. x , y >. | x F y } = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
13	3, 12	eqtrd 2262	. 2 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
14		df-mpt 4147	. 2 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
15	13, 14	eqtr4di 2280	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4083   {copab 4144   |-> cmpt 4145   Rel wrel 4724   Fn wfn 5313   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  fnrnfv  5680  feqmptd  5687  dffn5imf  5689  eqfnfv  5732  fndmin  5742  fcompt  5805  funiun  5816  resfunexg  5860  eufnfv  5870  fnovim  6113  offveqb  6238  caofinvl  6244  oprabco  6363  df1st2  6365  df2nd2  6366  pw2f1odclem  6995  xpen  7006  prdsbascl  13322  prdsidlem  13480  pws0g  13484  prdsinvlem  13641  cnmpt1st  14962  cnmpt2nd  14963