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Theorem ecinxp 8732
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 486 . . . . . . . 8 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → 𝐵𝐴)
21snssd 4770 . . . . . . 7 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
3 df-ss 3928 . . . . . . 7 ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
42, 3sylib 217 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ({𝐵} ∩ 𝐴) = {𝐵})
54imaeq2d 6014 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ ({𝐵} ∩ 𝐴)) = (𝑅 “ {𝐵}))
65ineq1d 4172 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) = ((𝑅 “ {𝐵}) ∩ 𝐴))
7 imass2 6055 . . . . . . 7 ({𝐵} ⊆ 𝐴 → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
82, 7syl 17 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
9 simpl 484 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅𝐴) ⊆ 𝐴)
108, 9sstrd 3955 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ 𝐴)
11 df-ss 3928 . . . . 5 ((𝑅 “ {𝐵}) ⊆ 𝐴 ↔ ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
1210, 11sylib 217 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
136, 12eqtr2d 2778 . . 3 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴))
14 imainrect 6134 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴)
1513, 14eqtr4di 2795 . 2 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}))
16 df-ec 8651 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
17 df-ec 8651 . 2 [𝐵](𝑅 ∩ (𝐴 × 𝐴)) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵})
1815, 16, 173eqtr4g 2802 1 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cin 3910  wss 3911  {csn 4587   × cxp 5632  cima 5637  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651
This theorem is referenced by:  qsinxp  8733  qusin  17427  pi1addval  24414  pi1grplem  24415
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