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

Theorem qsinxp 8812
Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Assertion
Ref Expression
qsinxp ((𝑅𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))

Proof of Theorem qsinxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecinxp 8811 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝑥𝐴) → [𝑥]𝑅 = [𝑥](𝑅 ∩ (𝐴 × 𝐴)))
21eqeq2d 2739 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝑥𝐴) → (𝑦 = [𝑥]𝑅𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
32rexbidva 3173 . . 3 ((𝑅𝐴) ⊆ 𝐴 → (∃𝑥𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
43abbidv 2797 . 2 ((𝑅𝐴) ⊆ 𝐴 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))})
5 df-qs 8731 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
6 df-qs 8731 . 2 (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))}
74, 5, 63eqtr4g 2793 1 ((𝑅𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {cab 2705  wrex 3067  cin 3946  wss 3947   × cxp 5676  cima 5681  [cec 8723   / cqs 8724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-qs 8731
This theorem is referenced by:  pi1buni  24980  pi1bas3  24983
  Copyright terms: Public domain W3C validator