Theorem fnsnsplitss 5510

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 3589	. . . 4 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) C_ A )
2	1	adantl 272	. . 3 |- ( ( F Fn A /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) C_ A )
3		ssres2 4753	. . 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 4739	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
6		fnressn 5497	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
7	6	uneq2d 3155	. . 3 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
8	5, 7	syl5eq 2133	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9		fnresdm 5136	. . 3 |- ( F Fn A -> ( F |` A ) = F )
10	9	adantr 271	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` A ) = F )
11	4, 8, 10	3sstr3d 3069	1 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   \ cdif 2997   u. cun 2998   C_ wss 3000   {csn 3450   <.cop 3453   |` cres 4454   Fn wfn 5023   ` cfv 5028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036

This theorem is referenced by:  funresdfunsnss  5514