Theorem fnsnsplitdc 6563

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 5367	. . 3 |- ( F Fn A -> ( F |` A ) = F )
2	1	3ad2ant2 1021	. 2 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( F |` A ) = F )
3		resundi 4959	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
4		dcdifsnid 6562	. . . . 5 |- ( ( A. x e. A A. y e. A DECID x = y /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) = A )
5	4	3adant2 1018	. . . 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 4946	. . 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 5748	. . . . 5 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
8	7	uneq2d 3317	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9	8	3adant1 1017	. . 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 2253	. 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 2231	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 835   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   \ cdif 3154   u. cun 3155   {csn 3622   <.cop 3625   |` cres 4665   Fn wfn 5253   ` cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266

This theorem is referenced by:  funresdfunsndc  6564