Theorem dffn5im 5624

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 5309 and dmmptss 5179. (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 5372	. . . 4 |- ( F Fn A -> Rel F )
2		dfrel4v 5134	. . . 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 5378	. . . . . . 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 2207	. . . . . . 7 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
8		fnbrfvb 5619	. . . . . . 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 4110	. . 3 |- ( F Fn A -> { <. x , y >. | x F y } = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
13	3, 12	eqtrd 2238	. 2 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
14		df-mpt 4107	. 2 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
15	13, 14	eqtr4di 2256	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   class class class wbr 4044   {copab 4104   |-> cmpt 4105   Rel wrel 4680   Fn wfn 5266   ` cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279

This theorem is referenced by:  fnrnfv  5625  feqmptd  5632  dffn5imf  5634  eqfnfv  5677  fndmin  5687  fcompt  5750  funiun  5761  resfunexg  5805  eufnfv  5815  fnovim  6054  offveqb  6178  caofinvl  6184  oprabco  6303  df1st2  6305  df2nd2  6306  pw2f1odclem  6931  xpen  6942  prdsbascl  13121  prdsidlem  13279  pws0g  13283  prdsinvlem  13440  cnmpt1st  14760  cnmpt2nd  14761