Theorem fnressn 5875

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 3705	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2	1	reseq2d 5043	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3		fveq2 5675	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
4		opeq12 3890	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ (𝐹‘𝑥) = (𝐹‘𝐵)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
5	3, 4	mpdan 421	. . . . . 6 ⊢ (𝑥 = 𝐵 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
6	5	sneqd 3707	. . . . 5 ⊢ (𝑥 = 𝐵 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝐵, (𝐹‘𝐵)⟩})
7	2, 6	eqeq12d 2249	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩} ↔ (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}))
8	7	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}) ↔ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩})))
9		vex 2818	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	snss 3834	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
11		fnssres 5476	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
12	10, 11	sylan2b 287	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13		dffn2 5515	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
14	9	fsn2 5856	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
15	13, 14	bitri 184	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
16		snssi 3843	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
17	16, 11	sylan2 286	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
18		vsnid 3726	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
19		funfvex 5692	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom (𝐹 ↾ {𝑥})) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
20	19	funfni 5463	. . . . . . . . 9 ⊢ (((𝐹 ↾ {𝑥}) Fn {𝑥} ∧ 𝑥 ∈ {𝑥}) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
21	17, 18, 20	sylancl 413	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
22	21	biantrurd 305	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
23		fvres 5699	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
24	18, 23	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
25	24	opeq2i 3892	. . . . . . . . 9 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
26	25	sneqi 3706	. . . . . . . 8 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
27	26	eqeq2i 2245	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
28	22, 27	bitr3di 195	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}))
29	15, 28	bitrid 192	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}))
30	12, 29	mpbid 147	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
31	30	expcom 116	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩}))
32	8, 31	vtoclga 2883	. 2 ⊢ (𝐵 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩}))
33	32	impcom 125	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  {csn 3694  ⟨cop 3697   ↾ cres 4756   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  fressnfv  5876  fnsnsplitss  5888  fnsnsplitdc  6751  dif1en  7149  fnfi  7216  fseq1p1m1  10450  resunimafz0  11223  trlsegvdegfi  16588  eupth2lem3lem2fi  16590  eupth2lem3lem6fi  16592  eupth2lem3lem4fi  16594