Theorem fnsnsplitdc 6668

Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)

Assertion

Ref	Expression
fnsnsplitdc	|- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> F = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )

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

Allowed substitution hints:   F( x, y)

Proof of Theorem fnsnsplitdc

Step	Hyp	Ref	Expression
1		fnresdm 5438	. . 3 |- ( F Fn A -> ( F |` A ) = F )
2	1	3ad2ant2 1043	. 2 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( F |` A ) = F )
3		resundi 5024	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
4		dcdifsnid 6667	. . . . 5 |- ( ( A. x e. A A. y e. A DECID x = y /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) = A )
5	4	3adant2 1040	. . . 4 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) = A )
6	5	reseq2d 5011	. . 3 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( F |` ( ( A \ { X } ) u. { X } ) ) = ( F |` A ) )
7		fnressn 5835	. . . . 5 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
8	7	uneq2d 3359	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9	8	3adant1 1039	. . 3 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
10	3, 6, 9	3eqtr3a 2286	. 2 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( F |` A ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
11	2, 10	eqtr3d 2264	1 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> F = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   \ cdif 3195   u. cun 3196   {csn 3667   <.cop 3670   |` cres 4725   Fn wfn 5319   ` cfv 5324

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332

This theorem is referenced by:  funresdfunsndc  6669