Theorem fnressn 5875

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 3705	. . . . . 6 |- ( x = B -> { x } = { B } )
2	1	reseq2d 5043	. . . . 5 |- ( x = B -> ( F |` { x } ) = ( F |` { B } ) )
3		fveq2 5675	. . . . . . 7 |- ( x = B -> ( F ` x ) = ( F ` B ) )
4		opeq12 3890	. . . . . . 7 |- ( ( x = B /\ ( F ` x ) = ( F ` B ) ) -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
5	3, 4	mpdan 421	. . . . . 6 |- ( x = B -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
6	5	sneqd 3707	. . . . 5 |- ( x = B -> { <. x , ( F ` x ) >. } = { <. B , ( F ` B ) >. } )
7	2, 6	eqeq12d 2249	. . . 4 |- ( x = B -> ( ( F |` { x } ) = { <. x , ( F ` x ) >. } <-> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
8	7	imbi2d 230	. . 3 |- ( x = B -> ( ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) <-> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) ) )
9		vex 2818	. . . . . . 7 |- x e. _V
10	9	snss 3834	. . . . . 6 |- ( x e. A <-> { x } C_ A )
11		fnssres 5476	. . . . . 6 |- ( ( F Fn A /\ { x } C_ A ) -> ( F |` { x } ) Fn { x } )
12	10, 11	sylan2b 287	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) Fn { x } )
13		dffn2 5515	. . . . . . 7 |- ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) : { x } --> _V )
14	9	fsn2 5856	. . . . . . 7 |- ( ( F |` { x } ) : { x } --> _V <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
15	13, 14	bitri 184	. . . . . 6 |- ( ( F |` { x } ) Fn { x } <-> ( ( ( F |` { x } ) ` x ) e. _V /\ ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } ) )
16		snssi 3843	. . . . . . . . . 10 |- ( x e. A -> { x } C_ A )
17	16, 11	sylan2 286	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) Fn { x } )
18		vsnid 3726	. . . . . . . . 9 |- x e. { x }
19		funfvex 5692	. . . . . . . . . 10 |- ( ( Fun ( F |` { x } ) /\ x e. dom ( F |` { x } ) ) -> ( ( F |` { x } ) ` x ) e. _V )
20	19	funfni 5463	. . . . . . . . 9 |- ( ( ( F |` { x } ) Fn { x } /\ x e. { x } ) -> ( ( F |` { x } ) ` x ) e. _V )
21	17, 18, 20	sylancl 413	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( ( F |` { x } ) ` x ) e. _V )
22	21	biantrurd 305	. . . . . . 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 5699	. . . . . . . . . . 11 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
24	18, 23	ax-mp 5	. . . . . . . . . 10 |- ( ( F |` { x } ) ` x ) = ( F ` x )
25	24	opeq2i 3892	. . . . . . . . 9 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
26	25	sneqi 3706	. . . . . . . 8 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
27	26	eqeq2i 2245	. . . . . . 7 |- ( ( F |` { x } ) = { <. x , ( ( F |` { x } ) ` x ) >. } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
28	22, 27	bitr3di 195	. . . . . 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	bitrid 192	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) )
30	12, 29	mpbid 147	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } )
31	30	expcom 116	. . 3 |- ( x e. A -> ( F Fn A -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) )
32	8, 31	vtoclga 2883	. 2 |- ( B e. A -> ( F Fn A -> ( F |` { B } ) = { <. B , ( F ` B ) >. } ) )
33	32	impcom 125	1 |- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   {csn 3694   <.cop 3697   |` cres 4756   Fn wfn 5352   -->wf 5353   ` cfv 5357

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  fressnfv  5876  fnsnsplitss  5888  fnsnsplitdc  6751  dif1en  7149  fnfi  7216  fseq1p1m1  10450  resunimafz0  11223  trlsegvdegfi  16588  eupth2lem3lem2fi  16590  eupth2lem3lem6fi  16592  eupth2lem3lem4fi  16594