Theorem fressnfv 7114

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 4595	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2		reseq2 5934	. . . . . . . 8 ⊢ ({𝑥} = {𝐵} → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3	2	feq1d 6652	. . . . . . 7 ⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝑥}⟶𝐶))
4		feq2 6649	. . . . . . 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 6840	. . . . . 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 7112	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
12		vsnid 4623	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥}
13		fvres 6859	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥))
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)
15	14	opeq2i 4837	. . . . . . . 8 ⊢ ⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩
16	15	sneqi 4596	. . . . . . 7 ⊢ {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹‘𝑥)⟩}
17	16	eqeq2i 2742	. . . . . 6 ⊢ ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
18		vex 3448	. . . . . . . 8 ⊢ 𝑥 ∈ V
19	18	fsn2 7090	. . . . . . 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 235	. . . . 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 3540	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4585  ⟨cop 4591   ↾ cres 5633   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by:  dif1enlem  9097  dif1enlemOLD  9098