Theorem fnsnsplitss 5852

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 3819	. . . 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 5040	. . 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 5026	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
6		fnressn 5839	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
7	6	uneq2d 3361	. . 3 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
8	5, 7	eqtrid 2276	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9		fnresdm 5441	. . 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 3271	1 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   \ cdif 3197   u. cun 3198   C_ wss 3200   {csn 3669   <.cop 3672   |` cres 4727   Fn wfn 5321   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  funresdfunsnss  5856