Theorem fnressn 2897

Description: A function restricted to a singleton.

Assertion

Ref	Expression
fnressn	⊢ ((F Fn A ∧ B ∈ A) → (F ↾ {B}) = {⟨B, (F ‘B)⟩})

Proof of Theorem fnressn

Step	Hyp	Ref	Expression
1		sneq 1816	. . . . . . 7 ⊢ (x = B → {x} = {B})
2		reseq2 2576	. . . . . . 7 ⊢ ({x} = {B} → (F ↾ {x}) = (F ↾ {B}))
3	1, 2	syl 12	. . . . . 6 ⊢ (x = B → (F ↾ {x}) = (F ↾ {B}))
4		fveq2 2832	. . . . . . . 8 ⊢ (x = B → (F ‘x) = (F ‘B))
5		opeq12 1878	. . . . . . . 8 ⊢ ((x = B ∧ (F ‘x) = (F ‘B)) → ⟨x, (F ‘x)⟩ = ⟨B, (F ‘B)⟩)
6	4, 5	mpdan 527	. . . . . . 7 ⊢ (x = B → ⟨x, (F ‘x)⟩ = ⟨B, (F ‘B)⟩)
7	6	sneqd 1818	. . . . . 6 ⊢ (x = B → {⟨x, (F ‘x)⟩} = {⟨B, (F ‘B)⟩})
8	3, 7	cleq12d 1115	. . . . 5 ⊢ (x = B → ((F ↾ {x}) = {⟨x, (F ‘x)⟩} ↔ (F ↾ {B}) = {⟨B, (F ‘B)⟩}))
9	8	imbi2d 464	. . . 4 ⊢ (x = B → ((F Fn A → (F ↾ {x}) = {⟨x, (F ‘x)⟩}) ↔ (F Fn A → (F ↾ {B}) = {⟨B, (F ‘B)⟩})))
10		fnssres 2734	. . . . . . . 8 ⊢ ((F Fn A ∧ {x} ⊆ A) → (F ↾ {x}) Fn {x})
11		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
12	11	snss 1849	. . . . . . . 8 ⊢ (x ∈ A ↔ {x} ⊆ A)
13	10, 12	sylan2b 347	. . . . . . 7 ⊢ ((F Fn A ∧ x ∈ A) → (F ↾ {x}) Fn {x})
14		fnf 2753	. . . . . . . 8 ⊢ ((F ↾ {x}) Fn {x} ↔ (F ↾ {x}):{x}–→V)
15	11	fsn2 2896	. . . . . . . 8 ⊢ ((F ↾ {x}):{x}–→V ↔ (((F ↾ {x}) ‘x) ∈ V ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}))
16		fvex 2838	. . . . . . . . . 10 ⊢ ((F ↾ {x}) ‘x) ∈ V
17	16	biantrur 544	. . . . . . . . 9 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} ↔ (((F ↾ {x}) ‘x) ∈ V ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}))
18	11	snid 1830	. . . . . . . . . . . . 13 ⊢ x ∈ {x}
19		fvres 2840	. . . . . . . . . . . . 13 ⊢ (x ∈ {x} → ((F ↾ {x}) ‘x) = (F ‘x))
20	18, 19	ax-mp 6	. . . . . . . . . . . 12 ⊢ ((F ↾ {x}) ‘x) = (F ‘x)
21		opeq2 1877	. . . . . . . . . . . 12 ⊢ (((F ↾ {x}) ‘x) = (F ‘x) → ⟨x, ((F ↾ {x}) ‘x)⟩ = ⟨x, (F ‘x)⟩)
22	20, 21	ax-mp 6	. . . . . . . . . . 11 ⊢ ⟨x, ((F ↾ {x}) ‘x)⟩ = ⟨x, (F ‘x)⟩
23	22	sneqi 1817	. . . . . . . . . 10 ⊢ {⟨x, ((F ↾ {x}) ‘x)⟩} = {⟨x, (F ‘x)⟩}
24	23	cleq2i 1111	. . . . . . . . 9 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} ↔ (F ↾ {x}) = {⟨x, (F ‘x)⟩})
25	17, 24	bitr3 153	. . . . . . . 8 ⊢ ((((F ↾ {x}) ‘x) ∈ V ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}) ↔ (F ↾ {x}) = {⟨x, (F ‘x)⟩})
26	14, 15, 25	3bitr 155	. . . . . . 7 ⊢ ((F ↾ {x}) Fn {x} ↔ (F ↾ {x}) = {⟨x, (F ‘x)⟩})
27	13, 26	sylib 173	. . . . . 6 ⊢ ((F Fn A ∧ x ∈ A) → (F ↾ {x}) = {⟨x, (F ‘x)⟩})
28	27	ancoms 334	. . . . 5 ⊢ ((x ∈ A ∧ F Fn A) → (F ↾ {x}) = {⟨x, (F ‘x)⟩})
29	28	exp 291	. . . 4 ⊢ (x ∈ A → (F Fn A → (F ↾ {x}) = {⟨x, (F ‘x)⟩}))
30	9, 29	vtoclga 1387	. . 3 ⊢ (B ∈ A → (F Fn A → (F ↾ {B}) = {⟨B, (F ‘B)⟩}))
31	30	imp 277	. 2 ⊢ ((B ∈ A ∧ F Fn A) → (F ↾ {B}) = {⟨B, (F ‘B)⟩})
32	31	ancoms 334	1 ⊢ ((F Fn A ∧ B ∈ A) → (F ↾ {B}) = {⟨B, (F ‘B)⟩})

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  {csn 1808  ⟨cop 1810   ↾ cres 2412   Fn wfn 2417  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  fressnfv 2898

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438

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