Theorem fnressn 5292

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 3378	. . . . . 6 |- (x = B -> {x } = {B })
2	1	reseq2d 4555	. . . . 5 |- (x = B -> ( F |` {x }) = ( F |` {B }))
3		fveq2 5121	. . . . . . 7 |- (x = B -> ( F ` x) = ( F ` B))
4		opeq12 3542	. . . . . . 7 |- ((x = B /\ ( F ` x) = ( F ` B)) -> <.x , ( F ` x) >. = <.B , ( F ` B) >.)
5	3, 4	mpdan 398	. . . . . 6 |- (x = B -> <.x , ( F ` x) >. = <.B , ( F ` B) >.)
6	5	sneqd 3380	. . . . 5 |- (x = B -> { <.x , ( F ` x) >. } = { <.B , ( F ` B) >. })
7	2, 6	eqeq12d 2051	. . . 4 |- (x = B -> (( F |` {x }) = { <.x , ( F ` x) >. } <-> ( F |` {B }) = { <.B , ( F ` B) >. }))
8	7	imbi2d 219	. . 3 |- (x = B -> (( F Fn A -> ( F |` {x }) = { <.x , ( F ` x) >. }) <-> ( F Fn A -> ( F |` {B }) = { <.B , ( F ` B) >. })))
9		vex 2554	. . . . . . 7 |- x e. _V
10	9	snss 3485	. . . . . 6 |- (x e. A <-> {x } C_ A)
11		fnssres 4955	. . . . . 6 |- (( F Fn A /\ {x } C_ A) -> ( F |` {x }) Fn {x })
12	10, 11	sylan2b 271	. . . . 5 |- (( F Fn A /\ x e. A) -> ( F |` {x }) Fn {x })
13		dffn2 4990	. . . . . . 7 |- (( F |` {x }) Fn {x } <-> ( F |` {x }) : {x } --> _V)
14	9	fsn2 5280	. . . . . . 7 |- (( F |` {x }) : {x } --> _V <-> ((( F |` {x }) ` x) e. _V /\ ( F |` {x }) = { <.x , (( F |` {x }) ` x) >. }))
15	13, 14	bitri 173	. . . . . 6 |- (( F |` {x }) Fn {x } <-> ((( F |` {x }) ` x) e. _V /\ ( F |` {x }) = { <.x , (( F |` {x }) ` x) >. }))
16		ssnid 3395	. . . . . . . . . . 11 |- x e. {x }
17		fvres 5141	. . . . . . . . . . 11 |- (x e. {x } -> (( F |` {x }) ` x) = ( F ` x))
18	16, 17	ax-mp 7	. . . . . . . . . 10 |- (( F |` {x }) ` x) = ( F ` x)
19	18	opeq2i 3544	. . . . . . . . 9 |- <.x , (( F |` {x }) ` x) >. = <.x , ( F ` x) >.
20	19	sneqi 3379	. . . . . . . 8 |- { <.x , (( F |` {x }) ` x) >. } = { <.x , ( F ` x) >. }
21	20	eqeq2i 2047	. . . . . . 7 |- (( F |` {x }) = { <.x , (( F |` {x }) ` x) >. } <-> ( F |` {x }) = { <.x , ( F ` x) >. })
22		snssi 3499	. . . . . . . . . 10 |- (x e. A -> {x } C_ A)
23	22, 11	sylan2 270	. . . . . . . . 9 |- (( F Fn A /\ x e. A) -> ( F |` {x }) Fn {x })
24		funfvex 5135	. . . . . . . . . 10 |- (( Fun ( F |` {x }) /\ x e. dom ( F |` {x })) -> (( F |` {x }) ` x) e. _V)
25	24	funfni 4942	. . . . . . . . 9 |- ((( F |` {x }) Fn {x } /\ x e. {x }) -> (( F |` {x }) ` x) e. _V)
26	23, 16, 25	sylancl 392	. . . . . . . 8 |- (( F Fn A /\ x e. A) -> (( F |` {x }) ` x) e. _V)
27	26	biantrurd 289	. . . . . . 7 |- (( F Fn A /\ x e. A) -> (( F |` {x }) = { <.x , (( F |` {x }) ` x) >. } <-> ((( F |` {x }) ` x) e. _V /\ ( F |` {x }) = { <.x , (( F |` {x }) ` x) >. })))
28	21, 27	syl5rbbr 184	. . . . . 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 181	. . . . 5 |- (( F Fn A /\ x e. A) -> (( F |` {x }) Fn {x } <-> ( F |` {x }) = { <.x , ( F ` x) >. }))
30	12, 29	mpbid 135	. . . 4 |- (( F Fn A /\ x e. A) -> ( F |` {x }) = { <.x , ( F ` x) >. })
31	30	expcom 109	. . 3 |- (x e. A -> ( F Fn A -> ( F |` {x }) = { <.x , ( F ` x) >. }))
32	8, 31	vtoclga 2613	. 2 |- (B e. A -> ( F Fn A -> ( F |` {B }) = { <.B , ( F ` B) >. }))
33	32	impcom 116	1 |- (( F Fn A /\ B e. A) -> ( F |` {B }) = { <.B , ( F ` B) >. })

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390   _Vcvv 2551   C_ wss 2911   {csn 3367   <.cop 3370   |` cres 4290   Fn wfn 4840   -->wf 4841   ` cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  fressnfv  5293  fseq1p1m1  8726