Theorem dffn5im 5745

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 5413 and dmmptss 5282. (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 5477	. . . 4 |- ( F Fn A -> Rel F )
2		dfrel4v 5237	. . . 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 5483	. . . . . . 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 398	. . . . 5 |- ( F Fn A -> ( x F y <-> ( x e. A /\ x F y ) ) )
7		eqcom 2240	. . . . . . 7 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
8		fnbrfvb 5738	. . . . . . 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 456	. . . . 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 4195	. . 3 |- ( F Fn A -> { <. x , y >. | x F y } = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
13	3, 12	eqtrd 2271	. 2 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
14		df-mpt 4192	. 2 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
15	13, 14	eqtr4di 2289	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4128   {copab 4189   |-> cmpt 4190   Rel wrel 4777   Fn wfn 5370   ` cfv 5375

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by:  fnrnfv  5746  feqmptd  5753  dffn5imf  5755  eqfnfv  5800  fndmin  5810  fcompt  5872  funiun  5884  resfunexg  5930  eufnfv  5942  fnovim  6190  offveqb  6315  caofinvl  6321  oprabco  6446  df1st2  6448  df2nd2  6449  pw2f1odclem  7127  xpen  7138  prdsbascl  14169  prdsidlem  14173  prdsinvlem  14176  pws0g  14193  cnmpt1st  15315  cnmpt2nd  15316