Theorem fnressn 5370

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 3409	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2	1	reseq2d 4630	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3		fveq2 5198	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
4		opeq12 3572	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ (𝐹‘𝑥) = (𝐹‘𝐵)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
5	3, 4	mpdan 412	. . . . . 6 ⊢ (𝑥 = 𝐵 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
6	5	sneqd 3411	. . . . 5 ⊢ (𝑥 = 𝐵 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝐵, (𝐹‘𝐵)⟩})
7	2, 6	eqeq12d 2095	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩} ↔ (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}))
8	7	imbi2d 228	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}) ↔ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩})))
9		vex 2604	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	snss 3516	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
11		fnssres 5032	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
12	10, 11	sylan2b 281	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13		dffn2 5067	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
14	9	fsn2 5358	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
15	13, 14	bitri 182	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
16		vsnid 3426	. . . . . . . . . . 11 ⊢ 𝑥 ∈ {𝑥}
17		fvres 5219	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
18	16, 17	ax-mp 7	. . . . . . . . . 10 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
19	18	opeq2i 3574	. . . . . . . . 9 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
20	19	sneqi 3410	. . . . . . . 8 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
21	20	eqeq2i 2091	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
22		snssi 3529	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
23	22, 11	sylan2 280	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
24		funfvex 5212	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom (𝐹 ↾ {𝑥})) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
25	24	funfni 5019	. . . . . . . . 9 ⊢ (((𝐹 ↾ {𝑥}) Fn {𝑥} ∧ 𝑥 ∈ {𝑥}) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
26	23, 16, 25	sylancl 404	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
27	26	biantrurd 299	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
28	21, 27	syl5rbbr 193	. . . . . 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 190	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}))
30	12, 29	mpbid 145	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
31	30	expcom 114	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}))
32	8, 31	vtoclga 2664	. 2 ⊢ (𝐵 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}))
33	32	impcom 123	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩})

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973  {csn 3398  ⟨cop 3401   ↾ cres 4365   Fn wfn 4917  ⟶wf 4918  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by:  fressnfv  5371  dif1en  6364  fnfi  6388  fseq1p1m1  9111