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

Theorem ecinxp 6347
Description: Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
Assertion
Ref Expression
ecinxp (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))

Proof of Theorem ecinxp
StepHypRef Expression
1 simpr 108 . . . . . . . 8 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → 𝐵𝐴)
21snssd 3577 . . . . . . 7 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
3 df-ss 3010 . . . . . . 7 ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
42, 3sylib 120 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ({𝐵} ∩ 𝐴) = {𝐵})
54imaeq2d 4761 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ ({𝐵} ∩ 𝐴)) = (𝑅 “ {𝐵}))
65ineq1d 3198 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) = ((𝑅 “ {𝐵}) ∩ 𝐴))
7 imass2 4795 . . . . . . 7 ({𝐵} ⊆ 𝐴 → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
82, 7syl 14 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
9 simpl 107 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅𝐴) ⊆ 𝐴)
108, 9sstrd 3033 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ 𝐴)
11 df-ss 3010 . . . . 5 ((𝑅 “ {𝐵}) ⊆ 𝐴 ↔ ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
1210, 11sylib 120 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
136, 12eqtr2d 2121 . . 3 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴))
14 imainrect 4863 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴)
1513, 14syl6eqr 2138 . 2 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}))
16 df-ec 6274 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
17 df-ec 6274 . 2 [𝐵](𝑅 ∩ (𝐴 × 𝐴)) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵})
1815, 16, 173eqtr4g 2145 1 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  cin 2996  wss 2997  {csn 3441   × cxp 4426  cima 4431  [cec 6270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-ec 6274
This theorem is referenced by:  qsinxp  6348  nqnq0pi  6976
  Copyright terms: Public domain W3C validator