Theorem fnressn 5439

Description: A function restricted to a singleton. (Contributed by set.mm contributors, 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 3745	. . . . . 6 ⊢ (x = B → {x} = {B})
2	1	reseq2d 4935	. . . . 5 ⊢ (x = B → (F ↾ {x}) = (F ↾ {B}))
3		fveq2 5329	. . . . . . 7 ⊢ (x = B → (F ‘x) = (F ‘B))
4		opeq12 4581	. . . . . . 7 ⊢ ((x = B ∧ (F ‘x) = (F ‘B)) → ⟨x, (F ‘x)⟩ = ⟨B, (F ‘B)⟩)
5	3, 4	mpdan 649	. . . . . 6 ⊢ (x = B → ⟨x, (F ‘x)⟩ = ⟨B, (F ‘B)⟩)
6	5	sneqd 3747	. . . . 5 ⊢ (x = B → {⟨x, (F ‘x)⟩} = {⟨B, (F ‘B)⟩})
7	2, 6	eqeq12d 2367	. . . 4 ⊢ (x = B → ((F ↾ {x}) = {⟨x, (F ‘x)⟩} ↔ (F ↾ {B}) = {⟨B, (F ‘B)⟩}))
8	7	imbi2d 307	. . 3 ⊢ (x = B → ((F Fn A → (F ↾ {x}) = {⟨x, (F ‘x)⟩}) ↔ (F Fn A → (F ↾ {B}) = {⟨B, (F ‘B)⟩})))
9		vex 2863	. . . . . . 7 ⊢ x ∈ V
10	9	snss 3839	. . . . . 6 ⊢ (x ∈ A ↔ {x} ⊆ A)
11		fnssres 5197	. . . . . 6 ⊢ ((F Fn A ∧ {x} ⊆ A) → (F ↾ {x}) Fn {x})
12	10, 11	sylan2b 461	. . . . 5 ⊢ ((F Fn A ∧ x ∈ A) → (F ↾ {x}) Fn {x})
13		dffn2 5225	. . . . . 6 ⊢ ((F ↾ {x}) Fn {x} ↔ (F ↾ {x}):{x}–→V)
14	9	fsn2 5435	. . . . . 6 ⊢ ((F ↾ {x}):{x}–→V ↔ (((F ↾ {x}) ‘x) ∈ V ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}))
15		fvex 5340	. . . . . . . 8 ⊢ ((F ↾ {x}) ‘x) ∈ V
16	15	biantrur 492	. . . . . . 7 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} ↔ (((F ↾ {x}) ‘x) ∈ V ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}))
17	9	snid 3761	. . . . . . . . . . 11 ⊢ x ∈ {x}
18		fvres 5343	. . . . . . . . . . 11 ⊢ (x ∈ {x} → ((F ↾ {x}) ‘x) = (F ‘x))
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ ((F ↾ {x}) ‘x) = (F ‘x)
20	19	opeq2i 4583	. . . . . . . . 9 ⊢ ⟨x, ((F ↾ {x}) ‘x)⟩ = ⟨x, (F ‘x)⟩
21	20	sneqi 3746	. . . . . . . 8 ⊢ {⟨x, ((F ↾ {x}) ‘x)⟩} = {⟨x, (F ‘x)⟩}
22	21	eqeq2i 2363	. . . . . . 7 ⊢ ((F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩} ↔ (F ↾ {x}) = {⟨x, (F ‘x)⟩})
23	16, 22	bitr3i 242	. . . . . 6 ⊢ ((((F ↾ {x}) ‘x) ∈ V ∧ (F ↾ {x}) = {⟨x, ((F ↾ {x}) ‘x)⟩}) ↔ (F ↾ {x}) = {⟨x, (F ‘x)⟩})
24	13, 14, 23	3bitri 262	. . . . 5 ⊢ ((F ↾ {x}) Fn {x} ↔ (F ↾ {x}) = {⟨x, (F ‘x)⟩})
25	12, 24	sylib 188	. . . 4 ⊢ ((F Fn A ∧ x ∈ A) → (F ↾ {x}) = {⟨x, (F ‘x)⟩})
26	25	expcom 424	. . 3 ⊢ (x ∈ A → (F Fn A → (F ↾ {x}) = {⟨x, (F ‘x)⟩}))
27	8, 26	vtoclga 2921	. 2 ⊢ (B ∈ A → (F Fn A → (F ↾ {B}) = {⟨B, (F ‘B)⟩}))
28	27	impcom 419	1 ⊢ ((F Fn A ∧ B ∈ A) → (F ↾ {B}) = {⟨B, (F ‘B)⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  {csn 3738  ⟨cop 4562   ↾ cres 4775   Fn wfn 4777  –→wf 4778   ‘cfv 4782

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796

This theorem is used by:  fressnfv  5440