Theorem funresdfunsndc 6717

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 5350	. . . . 5 |- ( Fun F -> Rel F )
2		resdmdfsn 5062	. . . . 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 1046	. . 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 3362	. 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 5363	. . 3 |- ( Fun F <-> F Fn dom F )
7		fnsnsplitdc 6716	. . 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 1311	. 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 2267	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 842   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   \ cdif 3198   u. cun 3199   {csn 3673   <.cop 3676   dom cdm 4731   |` cres 4733   Rel wrel 4736   Fun wfun 5327   Fn wfn 5328   ` cfv 5333

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 619  ax-in2 620  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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341

This theorem is referenced by:  strsetsid  13195