Theorem fnsnsplitss 5692

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 3724	. . . 4 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) C_ A )
2	1	adantl 275	. . 3 |- ( ( F Fn A /\ X e. A ) -> ( ( A \ { X } ) u. { X } ) C_ A )
3		ssres2 4916	. . 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 4902	. . 3 |- ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) )
6		fnressn 5679	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( F |` { X } ) = { <. X , ( F ` X ) >. } )
7	6	uneq2d 3281	. . 3 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. ( F |` { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
8	5, 7	eqtrid 2215	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` ( ( A \ { X } ) u. { X } ) ) = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
9		fnresdm 5305	. . 3 |- ( F Fn A -> ( F |` A ) = F )
10	9	adantr 274	. 2 |- ( ( F Fn A /\ X e. A ) -> ( F |` A ) = F )
11	4, 8, 10	3sstr3d 3191	1 |- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   \ cdif 3118   u. cun 3119   C_ wss 3121   {csn 3581   <.cop 3584   |` cres 4611   Fn wfn 5191   ` cfv 5196

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204

This theorem is referenced by:  funresdfunsnss  5696