Theorem fnressn 4812

Description: A function restricted to a singleton.

Assertion

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

Proof of Theorem fnressn

Step	Hyp	Ref	Expression
1		sneq 3054	. . . . . 6 |- (x = B -> {x} = {B})
2		reseq2 4219	. . . . . 6 |- ({x} = {B} -> (F |` {x}) = (F |` {B}))
3	1, 2	syl 12	. . . . 5 |- (x = B -> (F |` {x}) = (F |` {B}))
4		fveq2 4681	. . . . . . 7 |- (x = B -> (F` x) = (F` B))
5		opeq12 3160	. . . . . . 7 |- ((x = B /\ (F` x) = (F` B)) -> <.x, (F` x)>. = <.B, (F` B)>.)
6	4, 5	mpdan 768	. . . . . 6 |- (x = B -> <.x, (F` x)>. = <.B, (F` B)>.)
7	6	sneqd 3056	. . . . 5 |- (x = B -> {<.x, (F` x)>.} = {<.B, (F` B)>.})
8	3, 7	eqeq12d 1899	. . . 4 |- (x = B -> ((F |` {x}) = {<.x, (F` x)>.} <-> (F |` {B}) = {<.B, (F` B)>.}))
9	8	imbi2d 674	. . 3 |- (x = B -> ((F Fn A -> (F |` {x}) = {<.x, (F` x)>.}) <-> (F Fn A -> (F |` {B}) = {<.B, (F` B)>.})))
10		fnssres 4526	. . . . . 6 |- ((F Fn A /\ {x} C_ A) -> (F |` {x}) Fn {x})
11		visset 2295	. . . . . . 7 |- x e. _V
12	11	snss 3122	. . . . . 6 |- (x e. A <-> {x} C_ A)
13	10, 12	sylan2b 501	. . . . 5 |- ((F Fn A /\ x e. A) -> (F |` {x}) Fn {x})
14		dffn2 4563	. . . . . 6 |- ((F |` {x}) Fn {x} <-> (F |` {x}):{x}-->_V)
15	11	fsn2 4809	. . . . . 6 |- ((F |` {x}):{x}-->_V <-> (((F |` {x})` x) e. _V /\ (F |` {x}) = {<.x, ((F |` {x})` x)>.}))
16		fvex 4689	. . . . . . . 8 |- ((F |` {x})` x) e. _V
17	16	biantrur 794	. . . . . . 7 |- ((F |` {x}) = {<.x, ((F |` {x})` x)>.} <-> (((F |` {x})` x) e. _V /\ (F |` {x}) = {<.x, ((F |` {x})` x)>.}))
18	11	snid 3069	. . . . . . . . . . 11 |- x e. {x}
19		fvres 4691	. . . . . . . . . . 11 |- (x e. {x} -> ((F |` {x})` x) = (F` x))
20	18, 19	ax-mp 7	. . . . . . . . . 10 |- ((F |` {x})` x) = (F` x)
21	20	opeq2i 3162	. . . . . . . . 9 |- <.x, ((F |` {x})` x)>. = <.x, (F` x)>.
22	21	sneqi 3055	. . . . . . . 8 |- {<.x, ((F |` {x})` x)>.} = {<.x, (F` x)>.}
23	22	eqeq2i 1894	. . . . . . 7 |- ((F |` {x}) = {<.x, ((F |` {x})` x)>.} <-> (F |` {x}) = {<.x, (F` x)>.})
24	17, 23	bitr3i 192	. . . . . 6 |- ((((F |` {x})` x) e. _V /\ (F |` {x}) = {<.x, ((F |` {x})` x)>.}) <-> (F |` {x}) = {<.x, (F` x)>.})
25	14, 15, 24	3bitri 194	. . . . 5 |- ((F |` {x}) Fn {x} <-> (F |` {x}) = {<.x, (F` x)>.})
26	13, 25	sylib 215	. . . 4 |- ((F Fn A /\ x e. A) -> (F |` {x}) = {<.x, (F` x)>.})
27	26	expcom 403	. . 3 |- (x e. A -> (F Fn A -> (F |` {x}) = {<.x, (F` x)>.}))
28	9, 27	vtoclga 2352	. 2 |- (B e. A -> (F Fn A -> (F |` {B}) = {<.B, (F` B)>.}))
29	28	impcom 378	1 |- ((F Fn A /\ B e. A) -> (F |` {B}) = {<.B, (F` B)>.})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  {csn 3044  <.cop 3046   |` cres 3988   Fn wfn 3993  -->wf 3994  ` cfv 3998

This theorem is referenced by:  fressnfv 4813

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014

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