Theorem fnsnsplitdc 6488

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 5309	. . 3 |- ( F Fn A -> ( F |` A ) = F )
2	1	3ad2ant2 1015	. 2 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( F |` A ) = F )
3		resundi 4905	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
4		dcdifsnid 6487	. . . . 5 |- ( ( A. x e. A A. y e. A DECID x = y /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) = A )
5	4	3adant2 1012	. . . 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 4892	. . 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 5686	. . . . 5 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
8	7	uneq2d 3282	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9	8	3adant1 1011	. . 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 2228	. 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 2206	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 830   /\ w3a 974   = wceq 1349   e. wcel 2142   A.wral 2449   \ cdif 3119   u. cun 3120   {csn 3584   <.cop 3587   |` cres 4614   Fn wfn 5195   ` cfv 5200

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-dc 831  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-reu 2456  df-v 2733  df-sbc 2957  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-opab 4052  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208

This theorem is referenced by:  funresdfunsndc  6489