Theorem dffn5im 5729

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 5397 and dmmptss 5266. (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 5461	. . . 4 |- ( F Fn A -> Rel F )
2		dfrel4v 5221	. . . 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 5467	. . . . . . 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 2236	. . . . . . 7 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
8		fnbrfvb 5722	. . . . . . 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 4182	. . 3 |- ( F Fn A -> { <. x , y >. | x F y } = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
13	3, 12	eqtrd 2267	. 2 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
14		df-mpt 4179	. 2 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
15	13, 14	eqtr4di 2285	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4115   {copab 4176   |-> cmpt 4177   Rel wrel 4761   Fn wfn 5354   ` cfv 5359

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 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-iota 5319  df-fun 5361  df-fn 5362  df-fv 5367

This theorem is referenced by:  fnrnfv  5730  feqmptd  5737  dffn5imf  5739  eqfnfv  5782  fndmin  5792  fcompt  5854  funiun  5866  resfunexg  5912  eufnfv  5924  fnovim  6172  offveqb  6297  caofinvl  6303  oprabco  6428  df1st2  6430  df2nd2  6431  pw2f1odclem  7102  xpen  7113  prdsbascl  14138  prdsidlem  14142  prdsinvlem  14145  pws0g  14162  cnmpt1st  15284  cnmpt2nd  15285