Theorem fnressn 5671

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 3587	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2	1	reseq2d 4884	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3		fveq2 5486	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
4		opeq12 3760	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ (𝐹‘𝑥) = (𝐹‘𝐵)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
5	3, 4	mpdan 418	. . . . . 6 ⊢ (𝑥 = 𝐵 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
6	5	sneqd 3589	. . . . 5 ⊢ (𝑥 = 𝐵 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝐵, (𝐹‘𝐵)⟩})
7	2, 6	eqeq12d 2180	. . . 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 2729	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	snss 3702	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
11		fnssres 5301	. . . . . 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 5339	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
14	9	fsn2 5659	. . . . . . 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 3717	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
17	16, 11	sylan2 284	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
18		vsnid 3608	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
19		funfvex 5503	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom (𝐹 ↾ {𝑥})) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
20	19	funfni 5288	. . . . . . . . 9 ⊢ (((𝐹 ↾ {𝑥}) Fn {𝑥} ∧ 𝑥 ∈ {𝑥}) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
21	17, 18, 20	sylancl 410	. . . . . . . 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 5510	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
24	18, 23	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
25	24	opeq2i 3762	. . . . . . . . 9 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
26	25	sneqi 3588	. . . . . . . 8 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
27	26	eqeq2i 2176	. . . . . . 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 2792	. 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 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  {csn 3576  ⟨cop 3579   ↾ cres 4606   Fn wfn 5183  ⟶wf 5184  ‘cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  fressnfv  5672  fnsnsplitss  5684  fnsnsplitdc  6473  dif1en  6845  fnfi  6902  fseq1p1m1  10029  resunimafz0  10744