Theorem fnressn 5292

Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
fnressn	⊢ ((𝐹 Fn A ∧ B ∈ 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 ∈ V
10	9	snss 3485	. . . . . 6 ⊢ (x ∈ A ↔ {x} ⊆ A)
11		fnssres 4955	. . . . . 6 ⊢ ((𝐹 Fn A ∧ {x} ⊆ A) → (𝐹 ↾ {x}) Fn {x})
12	10, 11	sylan2b 271	. . . . 5 ⊢ ((𝐹 Fn A ∧ x ∈ 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) ∈ V ∧ (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}))
15	13, 14	bitri 173	. . . . . 6 ⊢ ((𝐹 ↾ {x}) Fn {x} ↔ (((𝐹 ↾ {x})‘x) ∈ V ∧ (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}))
16		ssnid 3395	. . . . . . . . . . 11 ⊢ x ∈ {x}
17		fvres 5141	. . . . . . . . . . 11 ⊢ (x ∈ {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 ∈ A → {x} ⊆ A)
23	22, 11	sylan2 270	. . . . . . . . 9 ⊢ ((𝐹 Fn A ∧ x ∈ A) → (𝐹 ↾ {x}) Fn {x})
24		funfvex 5135	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {x}) ∧ x ∈ dom (𝐹 ↾ {x})) → ((𝐹 ↾ {x})‘x) ∈ V)
25	24	funfni 4942	. . . . . . . . 9 ⊢ (((𝐹 ↾ {x}) Fn {x} ∧ x ∈ {x}) → ((𝐹 ↾ {x})‘x) ∈ V)
26	23, 16, 25	sylancl 392	. . . . . . . 8 ⊢ ((𝐹 Fn A ∧ x ∈ A) → ((𝐹 ↾ {x})‘x) ∈ V)
27	26	biantrurd 289	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ x ∈ A) → ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} ↔ (((𝐹 ↾ {x})‘x) ∈ V ∧ (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩})))
28	21, 27	syl5rbbr 184	. . . . . 6 ⊢ ((𝐹 Fn A ∧ x ∈ A) → ((((𝐹 ↾ {x})‘x) ∈ V ∧ (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}) ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹‘x)⟩}))
29	15, 28	syl5bb 181	. . . . 5 ⊢ ((𝐹 Fn A ∧ x ∈ A) → ((𝐹 ↾ {x}) Fn {x} ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹‘x)⟩}))
30	12, 29	mpbid 135	. . . 4 ⊢ ((𝐹 Fn A ∧ x ∈ A) → (𝐹 ↾ {x}) = {⟨x, (𝐹‘x)⟩})
31	30	expcom 109	. . 3 ⊢ (x ∈ A → (𝐹 Fn A → (𝐹 ↾ {x}) = {⟨x, (𝐹‘x)⟩}))
32	8, 31	vtoclga 2613	. 2 ⊢ (B ∈ A → (𝐹 Fn A → (𝐹 ↾ {B}) = {⟨B, (𝐹‘B)⟩}))
33	32	impcom 116	1 ⊢ ((𝐹 Fn A ∧ B ∈ A) → (𝐹 ↾ {B}) = {⟨B, (𝐹‘B)⟩})

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ⊆ 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