Theorem fnsnsplitdc 6738

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 5467	. . 3 |- ( F Fn A -> ( F |` A ) = F )
2	1	3ad2ant2 1046	. 2 |- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> ( F |` A ) = F )
3		resundi 5051	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
4		dcdifsnid 6737	. . . . 5 |- ( ( A. x e. A A. y e. A DECID x = y /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) = A )
5	4	3adant2 1043	. . . 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 5038	. . 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 5870	. . . . 5 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
8	7	uneq2d 3373	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9	8	3adant1 1042	. . 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 2289	. 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 2267	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 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   \ cdif 3208   u. cun 3209   {csn 3689   <.cop 3692   |` cres 4751   Fn wfn 5347   ` cfv 5352

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by:  funresdfunsndc  6739