Theorem fressnfv 7163

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 4634	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2		reseq2 5974	. . . . . . . 8 ⊢ ({𝑥} = {𝐵} → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3	2	feq1d 6701	. . . . . . 7 ⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝑥}⟶𝐶))
4		feq2 6698	. . . . . . 7 ⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝐵}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
5	3, 4	bitrd 279	. . . . . 6 ⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
6	1, 5	syl 17	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
7		fveq2 6891	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
8	7	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))
9	6, 8	bibi12d 345	. . . 4 ⊢ (𝑥 = 𝐵 → (((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶) ↔ ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)))
10	9	imbi2d 340	. . 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 7161	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
12		vsnid 4661	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥}
13		fvres 6910	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
15	14	opeq2i 4873	. . . . . . . 8 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
16	15	sneqi 4635	. . . . . . 7 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
17	16	eqeq2i 2740	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
18		vex 3473	. . . . . . . 8 ⊢ 𝑥 ∈ V
19	18	fsn2 7139	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
20		iba 527	. . . . . . . 8 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
21	14	eleq1i 2819	. . . . . . . 8 ⊢ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)
22	20, 21	bitr3di 286	. . . . . . 7 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹‘𝑥) ∈ 𝐶))
23	19, 22	bitrid 283	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
24	17, 23	sylbir 234	. . . . 5 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
25	11, 24	syl 17	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
26	25	expcom 413	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)))
27	10, 26	vtoclga 3561	. 2 ⊢ (𝐵 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)))
28	27	impcom 407	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {csn 4624  ⟨cop 4630   ↾ cres 5674   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by:  dif1enlem  9170  dif1enlemOLD  9171