MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecinxp Structured version   Visualization version   GIF version

Theorem ecinxp 8781
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 484 . . . . . . . 8 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → 𝐵𝐴)
21snssd 4804 . . . . . . 7 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
3 df-ss 3957 . . . . . . 7 ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
42, 3sylib 217 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ({𝐵} ∩ 𝐴) = {𝐵})
54imaeq2d 6049 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ ({𝐵} ∩ 𝐴)) = (𝑅 “ {𝐵}))
65ineq1d 4203 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) = ((𝑅 “ {𝐵}) ∩ 𝐴))
7 imass2 6091 . . . . . . 7 ({𝐵} ⊆ 𝐴 → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
82, 7syl 17 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
9 simpl 482 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅𝐴) ⊆ 𝐴)
108, 9sstrd 3984 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ 𝐴)
11 df-ss 3957 . . . . 5 ((𝑅 “ {𝐵}) ⊆ 𝐴 ↔ ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
1210, 11sylib 217 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
136, 12eqtr2d 2765 . . 3 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴))
14 imainrect 6170 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴)
1513, 14eqtr4di 2782 . 2 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}))
16 df-ec 8700 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
17 df-ec 8700 . 2 [𝐵](𝑅 ∩ (𝐴 × 𝐴)) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵})
1815, 16, 173eqtr4g 2789 1 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cin 3939  wss 3940  {csn 4620   × cxp 5664  cima 5669  [cec 8696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8700
This theorem is referenced by:  qsinxp  8782  qusin  17486  pi1addval  24885  pi1grplem  24886
  Copyright terms: Public domain W3C validator