Theorem funresdfunsndc 6535

Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)

Assertion

Ref	Expression
funresdfunsndc	|- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = F )

Distinct variable groups:   x, F, y   x, X, y

Proof of Theorem funresdfunsndc

Step	Hyp	Ref	Expression
1		funrel 5255	. . . . 5 |- ( Fun F -> Rel F )
2		resdmdfsn 4971	. . . . 5 |- ( Rel F -> ( F |` ( _V \ { X } ) ) = ( F |` ( dom F \ { X } ) ) )
3	1, 2	syl 14	. . . 4 |- ( Fun F -> ( F |` ( _V \ { X } ) ) = ( F |` ( dom F \ { X } ) ) )
4	3	3ad2ant2 1021	. . 3 |- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ Fun F /\ X e. dom F ) -> ( F |` ( _V \ { X } ) ) = ( F |` ( dom F \ { X } ) ) )
5	4	uneq1d 3303	. 2 |- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
6		funfn 5268	. . 3 |- ( Fun F <-> F Fn dom F )
7		fnsnsplitdc 6534	. . 3 |- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ F Fn dom F /\ X e. dom F ) -> F = ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
8	6, 7	syl3an2b 1286	. 2 |- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ Fun F /\ X e. dom F ) -> F = ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9	5, 8	eqtr4d 2225	1 |- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = F )

Syntax hints:   -> wi 4  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   _Vcvv 2752   \ cdif 3141   u. cun 3142   {csn 3610   <.cop 3613   dom cdm 4647   |` cres 4649   Rel wrel 4652   Fun wfun 5232   Fn wfn 5233   ` cfv 5238

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246

This theorem is referenced by:  strsetsid  12556