Theorem fressnfv 2898

Description: The value of a function restricted to a singleton.

Assertion

Ref	Expression
fressnfv	⊢ ((F Fn A ∧ B ∈ A) → ((F ↾ {B}):{B}–→C ↔ (F ‘B) ∈ C))

Proof of Theorem fressnfv

Step	Hyp	Ref	Expression
1		sneq 1816	. . . . . . 7 ⊢ (x = B → {x} = {B})
2		reseq2 2576	. . . . . . . . 9 ⊢ ({x} = {B} → (F ↾ {x}) = (F ↾ {B}))
3		feq1 2748	. . . . . . . . 9 ⊢ ((F ↾ {x}) = (F ↾ {B}) → ((F ↾ {x}):{x}–→C ↔ (F ↾ {B}):{x}–→C))
4	2, 3	syl 12	. . . . . . . 8 ⊢ ({x} = {B} → ((F ↾ {x}):{x}–→C ↔ (F ↾ {B}):{x}–→C))
5		feq2 2749	. . . . . . . 8 ⊢ ({x} = {B} → ((F ↾ {B}):{x}–→C ↔ (F ↾ {B}):{B}–→C))
6	4, 5	bitrd 406	. . . . . . 7 ⊢ ({x} = {B} → ((F ↾ {x}):{x}–→C ↔ (F ↾ {B}):{B}–→C))
7	1, 6	syl 12	. . . . . 6 ⊢ (x = B → ((F ↾ {x}):{x}–→C ↔ (F ↾ {B}):{B}–→C))
8		fveq2 2832	. . . . . . 7 ⊢ (x = B → (F ‘x) = (F ‘B))
9	8	eleq1d 1155	. . . . . 6 ⊢ (x = B → ((F ‘x) ∈ C ↔ (F ‘B) ∈ C))
10	7, 9	bibi12d 477	. . . . 5 ⊢ (x = B → (((F ↾ {x}):{x}–→C ↔ (F ‘x) ∈ C) ↔ ((F ↾ {B}):{B}–→C ↔ (F ‘B) ∈ C)))
11	10	imbi2d 464	. . . 4 ⊢ (x = B → ((F Fn A → ((F ↾ {x}):{x}–→C ↔ (F ‘x) ∈ C)) ↔ (F Fn A → ((F ↾ {B}):{B}–→C ↔ (F ‘B) ∈ C))))
12		fnressn 2897	. . . . . . 7 ⊢ ((F Fn A ∧ x ∈ A) → (F ↾ {x}) = {⟨x, (F ‘x)⟩})
13		visset 1350	. . . . . . . . . . . . 13 ⊢ x ∈ V
14	13	snid 1830	. . . . . . . . . . . 12 ⊢ x ∈ {x}
15		fvres 2840	. . . . . . . . . . . 12 ⊢ (x ∈ {x} → ((F ↾ {x}) ‘x) = (F ‘x))
16	14, 15	ax-mp 6	. . . . . . . . . . 11 ⊢ ((F ↾ {x}) ‘x) = (F ‘x)
17		opeq2 1877	. . . . . . . . . . 11 ⊢ (((F ↾ {x}) ‘x) = (F ‘x) → ⟨x, ((F ↾ {x}) ‘x)⟩ = ⟨x, (F ‘x)⟩)
18	16, 17	ax-mp 6	. . . . . . . . . 10 ⊢ ⟨x, ((F ↾ {x}) ‘x)⟩ = ⟨x, (F ‘x)⟩
19	18	sneqi 1817	. . . . . . . . 9 ⊢ {⟨x, ((F ↾ {x}) ‘x)⟩} = {⟨x, (F ‘x)⟩}
20	19	cleq2i 1111	. . . . . . . 8 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} ↔ (F ↾ {x}) = {⟨x, (F ‘x)⟩})
21		iba 486	. . . . . . . . . 10 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} → (((F ↾ {x}) ‘x) ∈ C ↔ (((F ↾ {x}) ‘x) ∈ C ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩})))
22	16	eleq1i 1152	. . . . . . . . . 10 ⊢ (((F ↾ {x}) ‘x) ∈ C ↔ (F ‘x) ∈ C)
23	21, 22	syl5rbbr 413	. . . . . . . . 9 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} → ((((F ↾ {x}) ‘x) ∈ C ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}) ↔ (F ‘x) ∈ C))
24	13	fsn2 2896	. . . . . . . . 9 ⊢ ((F ↾ {x}):{x}–→C ↔ (((F ↾ {x}) ‘x) ∈ C ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}))
25	23, 24	syl5bb 410	. . . . . . . 8 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} → ((F ↾ {x}):{x}–→C ↔ (F ‘x) ∈ C))
26	20, 25	sylbir 176	. . . . . . 7 ⊢ ((F ↾ {x}) = {⟨x, (F ‘x)⟩} → ((F ↾ {x}):{x}–→C ↔ (F ‘x) ∈ C))
27	12, 26	syl 12	. . . . . 6 ⊢ ((F Fn A ∧ x ∈ A) → ((F ↾ {x}):{x}–→C ↔ (F ‘x) ∈ C))
28	27	ancoms 334	. . . . 5 ⊢ ((x ∈ A ∧ F Fn A) → ((F ↾ {x}):{x}–→C ↔ (F ‘x) ∈ C))
29	28	exp 291	. . . 4 ⊢ (x ∈ A → (F Fn A → ((F ↾ {x}):{x}–→C ↔ (F ‘x) ∈ C)))
30	11, 29	vtoclga 1387	. . 3 ⊢ (B ∈ A → (F Fn A → ((F ↾ {B}):{B}–→C ↔ (F ‘B) ∈ C)))
31	30	imp 277	. 2 ⊢ ((B ∈ A ∧ F Fn A) → ((F ↾ {B}):{B}–→C ↔ (F ‘B) ∈ C))
32	31	ancoms 334	1 ⊢ ((F Fn A ∧ B ∈ A) → ((F ↾ {B}):{B}–→C ↔ (F ‘B) ∈ C))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {csn 1808  ⟨cop 1810   ↾ cres 2412   Fn wfn 2417  –→wf 2418   ‘cfv 2422

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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