ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecinxp Structured version   GIF version

Theorem ecinxp 6117
Description: Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
Assertion
Ref Expression
ecinxp (((𝑅A) ⊆ A B A) → [B]𝑅 = [B](𝑅 ∩ (A × A)))

Proof of Theorem ecinxp
StepHypRef Expression
1 simpr 103 . . . . . . . 8 (((𝑅A) ⊆ A B A) → B A)
21snssd 3500 . . . . . . 7 (((𝑅A) ⊆ A B A) → {B} ⊆ A)
3 df-ss 2925 . . . . . . 7 ({B} ⊆ A ↔ ({B} ∩ A) = {B})
42, 3sylib 127 . . . . . 6 (((𝑅A) ⊆ A B A) → ({B} ∩ A) = {B})
54imaeq2d 4611 . . . . 5 (((𝑅A) ⊆ A B A) → (𝑅 “ ({B} ∩ A)) = (𝑅 “ {B}))
65ineq1d 3131 . . . 4 (((𝑅A) ⊆ A B A) → ((𝑅 “ ({B} ∩ A)) ∩ A) = ((𝑅 “ {B}) ∩ A))
7 imass2 4644 . . . . . . 7 ({B} ⊆ A → (𝑅 “ {B}) ⊆ (𝑅A))
82, 7syl 14 . . . . . 6 (((𝑅A) ⊆ A B A) → (𝑅 “ {B}) ⊆ (𝑅A))
9 simpl 102 . . . . . 6 (((𝑅A) ⊆ A B A) → (𝑅A) ⊆ A)
108, 9sstrd 2949 . . . . 5 (((𝑅A) ⊆ A B A) → (𝑅 “ {B}) ⊆ A)
11 df-ss 2925 . . . . 5 ((𝑅 “ {B}) ⊆ A ↔ ((𝑅 “ {B}) ∩ A) = (𝑅 “ {B}))
1210, 11sylib 127 . . . 4 (((𝑅A) ⊆ A B A) → ((𝑅 “ {B}) ∩ A) = (𝑅 “ {B}))
136, 12eqtr2d 2070 . . 3 (((𝑅A) ⊆ A B A) → (𝑅 “ {B}) = ((𝑅 “ ({B} ∩ A)) ∩ A))
14 imainrect 4709 . . 3 ((𝑅 ∩ (A × A)) “ {B}) = ((𝑅 “ ({B} ∩ A)) ∩ A)
1513, 14syl6eqr 2087 . 2 (((𝑅A) ⊆ A B A) → (𝑅 “ {B}) = ((𝑅 ∩ (A × A)) “ {B}))
16 df-ec 6044 . 2 [B]𝑅 = (𝑅 “ {B})
17 df-ec 6044 . 2 [B](𝑅 ∩ (A × A)) = ((𝑅 ∩ (A × A)) “ {B})
1815, 16, 173eqtr4g 2094 1 (((𝑅A) ⊆ A B A) → [B]𝑅 = [B](𝑅 ∩ (A × A)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cin 2910  wss 2911  {csn 3367   × cxp 4286  cima 4291  [cec 6040
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-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  df-ec 6044
This theorem is referenced by:  qsinxp  6118  nqnq0pi  6420
  Copyright terms: Public domain W3C validator