Theorem fnressn 5760

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 3643	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2	1	reseq2d 4956	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3		fveq2 5570	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
4		opeq12 3820	. . . . . . 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 3645	. . . . 5 ⊢ (𝑥 = 𝐵 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝐵, (𝐹‘𝐵)⟩})
7	2, 6	eqeq12d 2219	. . . 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 2774	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	snss 3767	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
11		fnssres 5383	. . . . . 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 5421	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
14	9	fsn2 5748	. . . . . . 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 3776	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
17	16, 11	sylan2 286	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
18		vsnid 3664	. . . . . . . . 9 ⊢ 𝑥 ∈ {𝑥}
19		funfvex 5587	. . . . . . . . . 10 ⊢ ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom (𝐹 ↾ {𝑥})) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
20	19	funfni 5370	. . . . . . . . 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 5594	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
24	18, 23	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
25	24	opeq2i 3822	. . . . . . . . 9 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
26	25	sneqi 3644	. . . . . . . 8 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
27	26	eqeq2i 2215	. . . . . . 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 2838	. 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 1372   ∈ wcel 2175  Vcvv 2771   ⊆ wss 3165  {csn 3632  ⟨cop 3635   ↾ cres 4675   Fn wfn 5263  ⟶wf 5264  ‘cfv 5268

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276

This theorem is referenced by:  fressnfv  5761  fnsnsplitss  5773  fnsnsplitdc  6581  dif1en  6958  fnfi  7020  fseq1p1m1  10198  resunimafz0  10957