Theorem fnressn 5682

Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
fnressn	|- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )

Proof of Theorem fnressn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3594	. . . . . 6 |- ( x = B -> { x } = { B } )
2	1	reseq2d 4891	. . . . 5 |- ( x = B -> ( F |` { x } ) = ( F |` { B } ) )
3		fveq2 5496	. . . . . . 7 |- ( x = B -> ( F ` x ) = ( F ` B ) )
4		opeq12 3767	. . . . . . 7 |- ( ( x = B /\ ( F ` x ) = ( F ` B ) ) -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
5	3, 4	mpdan 419	. . . . . 6 |- ( x = B -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
6	5	sneqd 3596	. . . . 5 |- ( x = B -> { <. x , ( F ` x ) >. } = { <. B , ( F ` B ) >. } )
7	2, 6	eqeq12d 2185	. . . 4 |- ( x = B -> ( ( F |` { x } ) = { <. x , ( F ` x ) >. } <-> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
8	7	imbi2d 229	. . 3 |- ( x = B -> ( ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) <-> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) ) )
9		vex 2733	. . . . . . 7 |- x e. _V
10	9	snss 3709	. . . . . 6 |- ( x e. A <-> { x } C_ A )
11		fnssres 5311	. . . . . 6 |- ( ( F Fn A /\ { x } C_ A ) -> ( F |` { x } ) Fn { x } )
12	10, 11	sylan2b 285	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) Fn { x } )
13		dffn2 5349	. . . . . . 7 |- ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) : { x } --> _V )
14	9	fsn2 5670	. . . . . . 7 |- ( ( F |` { x } ) : { x } --> _V <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
15	13, 14	bitri 183	. . . . . 6 |- ( ( F |` { x } ) Fn { x } <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
16		snssi 3724	. . . . . . . . . 10 |- ( x e. A -> { x } C_ A )
17	16, 11	sylan2 284	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) Fn { x } )
18		vsnid 3615	. . . . . . . . 9 |- x e. { x }
19		funfvex 5513	. . . . . . . . . 10 |- ( ( Fun ( F |` { x } ) /\ x e. dom ( F |` { x } ) ) -> ( ( F |` { x } ) ` x ) e. _V )
20	19	funfni 5298	. . . . . . . . 9 |- ( ( ( F |` { x } ) Fn { x } /\ x e. { x } ) -> ( ( F |` { x } ) ` x ) e. _V )
21	17, 18, 20	sylancl 411	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( ( F |` { x } ) ` x ) e. _V )
22	21	biantrurd 303	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) ) )
23		fvres 5520	. . . . . . . . . . 11 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
24	18, 23	ax-mp 5	. . . . . . . . . 10 |- ( ( F |` { x } ) ` x ) = ( F ` x )
25	24	opeq2i 3769	. . . . . . . . 9 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
26	25	sneqi 3595	. . . . . . . 8 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
27	26	eqeq2i 2181	. . . . . . 7 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
28	22, 27	bitr3di 194	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> ( ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) )
29	15, 28	syl5bb 191	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) )
30	12, 29	mpbid 146	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
31	30	expcom 115	. . 3 |- ( x e. A -> ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) )
32	8, 31	vtoclga 2796	. 2 |- ( B e. A -> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
33	32	impcom 124	1 |- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   {csn 3583   <.cop 3586   |` cres 4613   Fn wfn 5193   -->wf 5194   ` cfv 5198

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-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 4107  ax-pow 4160  ax-pr 4194

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-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by:  fressnfv  5683  fnsnsplitss  5695  fnsnsplitdc  6484  dif1en  6857  fnfi  6914  fseq1p1m1  10050  resunimafz0  10766