Theorem fnsnsplitss 5757

Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)

Assertion

Ref	Expression
fnsnsplitss	|- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Proof of Theorem fnsnsplitss

Step	Hyp	Ref	Expression
1		difsnss 3764	. . . 4 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) C_ A )
2	1	adantl 277	. . 3 |- ( ( F Fn A /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) C_ A )
3		ssres2 4969	. . 3 |- ( ( ( A \ { X } ) u. { X } ) C_ A -> ( F |` ( ( A \ { X } ) u. { X } ) ) C_ ( F |` A ) )
4	2, 3	syl 14	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` ( ( A \ { X } ) u. { X } ) ) C_ ( F |` A ) )
5		resundi 4955	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
6		fnressn 5744	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
7	6	uneq2d 3313	. . 3 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
8	5, 7	eqtrid 2238	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9		fnresdm 5363	. . 3 |- ( F Fn A -> ( F |` A ) = F )
10	9	adantr 276	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` A ) = F )
11	4, 8, 10	3sstr3d 3223	1 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   \ cdif 3150   u. cun 3151   C_ wss 3153   {csn 3618   <.cop 3621   |` cres 4661   Fn wfn 5249   ` cfv 5254

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262

This theorem is referenced by:  funresdfunsnss  5761