Theorem fressnfv 5683

Description: The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
fressnfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))

Proof of Theorem fressnfv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3594	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2		reseq2 4886	. . . . . . . 8 ⊢ ({𝑥} = {𝐵} → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3	2	feq1d 5334	. . . . . . 7 ⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝑥}⟶𝐶))
4		feq2 5331	. . . . . . 7 ⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝐵}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
5	3, 4	bitrd 187	. . . . . 6 ⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
6	1, 5	syl 14	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
7		fveq2 5496	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
8	7	eleq1d 2239	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))
9	6, 8	bibi12d 234	. . . 4 ⊢ (𝑥 = 𝐵 → (((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶) ↔ ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)))
10	9	imbi2d 229	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) ↔ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))))
11		fnressn 5682	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
12		vsnid 3615	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥}
13		fvres 5520	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
15	14	opeq2i 3769	. . . . . . . 8 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
16	15	sneqi 3595	. . . . . . 7 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
17	16	eqeq2i 2181	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
18		vex 2733	. . . . . . . 8 ⊢ 𝑥 ∈ V
19	18	fsn2 5670	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
20		iba 298	. . . . . . . 8 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
21	14	eleq1i 2236	. . . . . . . 8 ⊢ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)
22	20, 21	bitr3di 194	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹‘𝑥) ∈ 𝐶))
23	19, 22	syl5bb 191	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
24	17, 23	sylbir 134	. . . . 5 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
25	11, 24	syl 14	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
26	25	expcom 115	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)))
27	10, 26	vtoclga 2796	. 2 ⊢ (𝐵 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)))
28	27	impcom 124	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {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: (None)