Theorem fnressn 3837

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 2417	. . . . . 6 |- (x = B -> {x} = {B})
2		reseq2 3369	. . . . . 6 |- ({x} = {B} -> (F |` {x}) = (F |` {B}))
3	1, 2	syl 10	. . . . 5 |- (x = B -> (F |` {x}) = (F |` {B}))
4		fveq2 3724	. . . . . . 7 |- (x = B -> (F` x) = (F` B))
5		opeq12 2489	. . . . . . 7 |- ((x = B /\ (F` x) = (F` B)) -> <.x, (F` x)>. = <.B, (F` B)>.)
6	4, 5	mpdan 704	. . . . . 6 |- (x = B -> <.x, (F` x)>. = <.B, (F` B)>.)
7	6	sneqd 2419	. . . . 5 |- (x = B -> {<.x, (F` x)>.} = {<.B, (F` B)>.})
8	3, 7	eqeq12d 1489	. . . 4 |- (x = B -> ((F |` {x}) = {<.x, (F` x)>.} <-> (F |` {B}) = {<.B, (F` B)>.}))
9	8	imbi2d 612	. . 3 |- (x = B -> ((F Fn A -> (F |` {x}) = {<.x, (F` x)>.}) <-> (F Fn A -> (F |` {B}) = {<.B, (F` B)>.})))
10		fnssres 3600	. . . . . 6 |- ((F Fn A /\ {x} (_ A) -> (F |` {x}) Fn {x})
11		visset 1813	. . . . . . 7 |- x e. V
12	11	snss 2461	. . . . . 6 |- (x e. A <-> {x} (_ A)
13	10, 12	sylan2b 452	. . . . 5 |- ((F Fn A /\ x e. A) -> (F |` {x}) Fn {x})
14		fnf 3628	. . . . . 6 |- ((F |` {x}) Fn {x} <-> (F |` {x}):{x}-->V)
15	11	fsn2 3836	. . . . . 6 |- ((F |` {x}):{x}-->V <-> (((F |` {x})` x) e. V /\ (F |` {x}) = {<.x, ((F |` {x})` x)>.}))
16		fvex 3732	. . . . . . . 8 |- ((F |` {x})` x) e. V
17	16	biantrur 725	. . . . . . 7 |- ((F |` {x}) = {<.x, ((F |` {x})` x)>.} <-> (((F |` {x})` x) e. V /\ (F |` {x}) = {<.x, ((F |` {x})` x)>.}))
18	11	snid 2435	. . . . . . . . . . 11 |- x e. {x}
19		fvres 3734	. . . . . . . . . . 11 |- (x e. {x} -> ((F |` {x})` x) = (F` x))
20	18, 19	ax-mp 7	. . . . . . . . . 10 |- ((F |` {x})` x) = (F` x)
21	20	opeq2i 2491	. . . . . . . . 9 |- <.x, ((F |` {x})` x)>. = <.x, (F` x)>.
22	21	sneqi 2418	. . . . . . . 8 |- {<.x, ((F |` {x})` x)>.} = {<.x, (F` x)>.}
23	22	eqeq2i 1485	. . . . . . 7 |- ((F |` {x}) = {<.x, ((F |` {x})` x)>.} <-> (F |` {x}) = {<.x, (F` x)>.})
24	17, 23	bitr3 175	. . . . . 6 |- ((((F |` {x})` x) e. V /\ (F |` {x}) = {<.x, ((F |` {x})` x)>.}) <-> (F |` {x}) = {<.x, (F` x)>.})
25	14, 15, 24	3bitr 177	. . . . 5 |- ((F |` {x}) Fn {x} <-> (F |` {x}) = {<.x, (F` x)>.})
26	13, 25	sylib 198	. . . 4 |- ((F Fn A /\ x e. A) -> (F |` {x}) = {<.x, (F` x)>.})
27	26	expcom 374	. . 3 |- (x e. A -> (F Fn A -> (F |` {x}) = {<.x, (F` x)>.}))
28	9, 27	vtoclga 1852	. 2 |- (B e. A -> (F Fn A -> (F |` {B}) = {<.B, (F` B)>.}))
29	28	impcom 351	1 |- ((F Fn A /\ B e. A) -> (F |` {B}) = {<.B, (F` B)>.})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  {csn 2409  <.cop 2411   |` cres 3172   Fn wfn 3177  -->wf 3178  ` cfv 3182

This theorem is referenced by:  fressnfv 3838

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-reu 1651  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198

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