Theorem fnressn 5682

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 3594	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2	1	reseq2d 4891	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3		fveq2 5496	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
4		opeq12 3767	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ (𝐹‘𝑥) = (𝐹‘𝐵)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
5	3, 4	mpdan 419	. . . . . 6 ⊢ (𝑥 = 𝐵 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
6	5	sneqd 3596	. . . . 5 ⊢ (𝑥 = 𝐵 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝐵, (𝐹‘𝐵)⟩})
7	2, 6	eqeq12d 2185	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩} ↔ (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}))
8	7	imbi2d 229	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}) ↔ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩})))
9		vex 2733	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	snss 3709	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
11		fnssres 5311	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
12	10, 11	sylan2b 285	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13		dffn2 5349	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
14	9	fsn2 5670	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
15	13, 14	bitri 183	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
16		snssi 3724	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
17	16, 11	sylan2 284	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
18		vsnid 3615	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
19		funfvex 5513	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom (𝐹 ↾ {𝑥})) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
20	19	funfni 5298	. . . . . . . . 9 ⊢ (((𝐹 ↾ {𝑥}) Fn {𝑥} ∧ 𝑥 ∈ {𝑥}) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
21	17, 18, 20	sylancl 411	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
22	21	biantrurd 303	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
23		fvres 5520	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
24	18, 23	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
25	24	opeq2i 3769	. . . . . . . . 9 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
26	25	sneqi 3595	. . . . . . . 8 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
27	26	eqeq2i 2181	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
28	22, 27	bitr3di 194	. . . . . 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 191	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}))
30	12, 29	mpbid 146	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
31	30	expcom 115	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}))
32	8, 31	vtoclga 2796	. 2 ⊢ (𝐵 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}))
33	32	impcom 124	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  {csn 3583  ⟨cop 3586   ↾ cres 4613   Fn wfn 5193  ⟶wf 5194  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by:  fressnfv  5683  fnsnsplitss  5695  fnsnsplitdc  6484  dif1en  6857  fnfi  6914  fseq1p1m1  10050  resunimafz0  10766