Theorem fressnfv 5293

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

Assertion

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

Proof of Theorem fressnfv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3378	. . . . . 6 ⊢ (x = B → {x} = {B})
2		reseq2 4550	. . . . . . . 8 ⊢ ({x} = {B} → (𝐹 ↾ {x}) = (𝐹 ↾ {B}))
3	2	feq1d 4977	. . . . . . 7 ⊢ ({x} = {B} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{x}⟶𝐶))
4		feq2 4974	. . . . . . 7 ⊢ ({x} = {B} → ((𝐹 ↾ {B}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{B}⟶𝐶))
5	3, 4	bitrd 177	. . . . . 6 ⊢ ({x} = {B} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{B}⟶𝐶))
6	1, 5	syl 14	. . . . 5 ⊢ (x = B → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{B}⟶𝐶))
7		fveq2 5121	. . . . . 6 ⊢ (x = B → (𝐹‘x) = (𝐹‘B))
8	7	eleq1d 2103	. . . . 5 ⊢ (x = B → ((𝐹‘x) ∈ 𝐶 ↔ (𝐹‘B) ∈ 𝐶))
9	6, 8	bibi12d 224	. . . 4 ⊢ (x = B → (((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹‘x) ∈ 𝐶) ↔ ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹‘B) ∈ 𝐶)))
10	9	imbi2d 219	. . 3 ⊢ (x = B → ((𝐹 Fn A → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹‘x) ∈ 𝐶)) ↔ (𝐹 Fn A → ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹‘B) ∈ 𝐶))))
11		fnressn 5292	. . . . 5 ⊢ ((𝐹 Fn A ∧ x ∈ A) → (𝐹 ↾ {x}) = {⟨x, (𝐹‘x)⟩})
12		ssnid 3395	. . . . . . . . . 10 ⊢ x ∈ {x}
13		fvres 5141	. . . . . . . . . 10 ⊢ (x ∈ {x} → ((𝐹 ↾ {x})‘x) = (𝐹‘x))
14	12, 13	ax-mp 7	. . . . . . . . 9 ⊢ ((𝐹 ↾ {x})‘x) = (𝐹‘x)
15	14	opeq2i 3544	. . . . . . . 8 ⊢ ⟨x, ((𝐹 ↾ {x})‘x)⟩ = ⟨x, (𝐹‘x)⟩
16	15	sneqi 3379	. . . . . . 7 ⊢ {⟨x, ((𝐹 ↾ {x})‘x)⟩} = {⟨x, (𝐹‘x)⟩}
17	16	eqeq2i 2047	. . . . . 6 ⊢ ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹‘x)⟩})
18		vex 2554	. . . . . . . 8 ⊢ x ∈ V
19	18	fsn2 5280	. . . . . . 7 ⊢ ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (((𝐹 ↾ {x})‘x) ∈ 𝐶 ∧ (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}))
20	14	eleq1i 2100	. . . . . . . 8 ⊢ (((𝐹 ↾ {x})‘x) ∈ 𝐶 ↔ (𝐹‘x) ∈ 𝐶)
21		iba 284	. . . . . . . 8 ⊢ ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} → (((𝐹 ↾ {x})‘x) ∈ 𝐶 ↔ (((𝐹 ↾ {x})‘x) ∈ 𝐶 ∧ (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩})))
22	20, 21	syl5rbbr 184	. . . . . . 7 ⊢ ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} → ((((𝐹 ↾ {x})‘x) ∈ 𝐶 ∧ (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}) ↔ (𝐹‘x) ∈ 𝐶))
23	19, 22	syl5bb 181	. . . . . 6 ⊢ ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹‘x) ∈ 𝐶))
24	17, 23	sylbir 125	. . . . 5 ⊢ ((𝐹 ↾ {x}) = {⟨x, (𝐹‘x)⟩} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹‘x) ∈ 𝐶))
25	11, 24	syl 14	. . . 4 ⊢ ((𝐹 Fn A ∧ x ∈ A) → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹‘x) ∈ 𝐶))
26	25	expcom 109	. . 3 ⊢ (x ∈ A → (𝐹 Fn A → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹‘x) ∈ 𝐶)))
27	10, 26	vtoclga 2613	. 2 ⊢ (B ∈ A → (𝐹 Fn A → ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹‘B) ∈ 𝐶)))
28	27	impcom 116	1 ⊢ ((𝐹 Fn A ∧ B ∈ A) → ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹‘B) ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {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: (None)