Theorem fnressn 5904

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 3720	. . . . . 6 |- ( x = B -> { x } = { B } )
2	1	reseq2d 5065	. . . . 5 |- ( x = B -> ( F |` { x } ) = ( F |` { B } ) )
3		fveq2 5698	. . . . . . 7 |- ( x = B -> ( F ` x ) = ( F ` B ) )
4		opeq12 3906	. . . . . . 7 |- ( ( x = B /\ ( F ` x ) = ( F ` B ) ) -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
5	3, 4	mpdan 425	. . . . . 6 |- ( x = B -> <. x , ( F ` x ) >. = <. B , ( F ` B ) >. )
6	5	sneqd 3722	. . . . 5 |- ( x = B -> { <. x , ( F ` x ) >. } = { <. B , ( F ` B ) >. } )
7	2, 6	eqeq12d 2253	. . . 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 2824	. . . . . . 7 |- x e. _V
10	9	snss 3850	. . . . . 6 |- ( x e. A <-> { x } C_ A )
11		fnssres 5499	. . . . . 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 5538	. . . . . . 7 |- ( ( F |` { x } ) Fn { x } <-> ( F |` { x } ) : { x } --> _V )
14	9	fsn2 5885	. . . . . . 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 3859	. . . . . . . . . 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 3741	. . . . . . . . 9 |- x e. { x }
19		funfvex 5715	. . . . . . . . . 10 |- ( ( Fun ( F |` { x } ) /\ x e. dom ( F |` { x } ) ) -> ( ( F |` { x } ) ` x ) e. _V )
20	19	funfni 5486	. . . . . . . . 9 |- ( ( ( F |` { x } ) Fn { x } /\ x e. { x } ) -> ( ( F |` { x } ) ` x ) e. _V )
21	17, 18, 20	sylancl 417	. . . . . . . 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 5722	. . . . . . . . . . 11 |- ( x e. { x } -> ( ( F |` { x } ) ` x ) = ( F ` x ) )
24	18, 23	ax-mp 5	. . . . . . . . . 10 |- ( ( F |` { x } ) ` x ) = ( F ` x )
25	24	opeq2i 3908	. . . . . . . . 9 |- <. x , ( ( F |` { x } ) ` x ) >. = <. x , ( F ` x ) >.
26	25	sneqi 3721	. . . . . . . 8 |- { <. x , ( ( F |` { x } ) ` x ) >. } = { <. x , ( F ` x ) >. }
27	26	eqeq2i 2249	. . . . . . 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 2889	. 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 ) >. } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {csn 3709   <.cop 3712   |` cres 4778   Fn wfn 5375   -->wf 5376   ` cfv 5380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388

This theorem is used by:  fressnfv  5905  fnsnsplitss  5917  fnsnsplitdc  6782  dif1en  7187  fnfi  7254  fseq1p1m1  10518  resunimafz0  11296  trlsegvdegfi  16913  eupth2lem3lem2fi  16915  eupth2lem3lem6fi  16917  eupth2lem3lem4fi  16919