Theorem ressn 4801

Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Assertion

Ref	Expression
ressn	⊢ (A ↾ {B}) = ({B} × (A “ {B}))

Proof of Theorem ressn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4582	. 2 ⊢ Rel (A ↾ {B})
2		relxp 4390	. 2 ⊢ Rel ({B} × (A “ {B}))
3		ancom 253	. . . 4 ⊢ ((⟨x, y⟩ ∈ A ∧ x ∈ {B}) ↔ (x ∈ {B} ∧ ⟨x, y⟩ ∈ A))
4		vex 2554	. . . . . . 7 ⊢ x ∈ V
5		vex 2554	. . . . . . 7 ⊢ y ∈ V
6	4, 5	elimasn 4635	. . . . . 6 ⊢ (y ∈ (A “ {x}) ↔ ⟨x, y⟩ ∈ A)
7		elsni 3391	. . . . . . . . 9 ⊢ (x ∈ {B} → x = B)
8	7	sneqd 3380	. . . . . . . 8 ⊢ (x ∈ {B} → {x} = {B})
9	8	imaeq2d 4611	. . . . . . 7 ⊢ (x ∈ {B} → (A “ {x}) = (A “ {B}))
10	9	eleq2d 2104	. . . . . 6 ⊢ (x ∈ {B} → (y ∈ (A “ {x}) ↔ y ∈ (A “ {B})))
11	6, 10	syl5bbr 183	. . . . 5 ⊢ (x ∈ {B} → (⟨x, y⟩ ∈ A ↔ y ∈ (A “ {B})))
12	11	pm5.32i 427	. . . 4 ⊢ ((x ∈ {B} ∧ ⟨x, y⟩ ∈ A) ↔ (x ∈ {B} ∧ y ∈ (A “ {B})))
13	3, 12	bitri 173	. . 3 ⊢ ((⟨x, y⟩ ∈ A ∧ x ∈ {B}) ↔ (x ∈ {B} ∧ y ∈ (A “ {B})))
14	5	opelres 4560	. . 3 ⊢ (⟨x, y⟩ ∈ (A ↾ {B}) ↔ (⟨x, y⟩ ∈ A ∧ x ∈ {B}))
15		opelxp 4317	. . 3 ⊢ (⟨x, y⟩ ∈ ({B} × (A “ {B})) ↔ (x ∈ {B} ∧ y ∈ (A “ {B})))
16	13, 14, 15	3bitr4i 201	. 2 ⊢ (⟨x, y⟩ ∈ (A ↾ {B}) ↔ ⟨x, y⟩ ∈ ({B} × (A “ {B})))
17	1, 2, 16	eqrelriiv 4377	1 ⊢ (A ↾ {B}) = ({B} × (A “ {B}))

Syntax hints:   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {csn 3367  ⟨cop 3370   × cxp 4286   ↾ cres 4290   “ cima 4291

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by: (None)