Theorem fnsnsplitdc 6473

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 5297	. . 3 |- ( F Fn A -> ( F |` A ) = F )
2	1	3ad2ant2 1009	. 2 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( F |` A ) = F )
3		resundi 4897	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
4		dcdifsnid 6472	. . . . 5 |- ( ( A. x e. A A. y e. A DECID x = y /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) = A )
5	4	3adant2 1006	. . . 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 4884	. . 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 5671	. . . . 5 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
8	7	uneq2d 3276	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9	8	3adant1 1005	. . 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 2223	. 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 2200	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 103  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   \ cdif 3113   u. cun 3114   {csn 3576   <.cop 3579   |` cres 4606   Fn wfn 5183   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  funresdfunsndc  6474