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Theorem ecinxp 6472
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 109 . . . . . . . 8 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → 𝐵𝐴)
21snssd 3635 . . . . . . 7 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
3 df-ss 3054 . . . . . . 7 ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
42, 3sylib 121 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ({𝐵} ∩ 𝐴) = {𝐵})
54imaeq2d 4851 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ ({𝐵} ∩ 𝐴)) = (𝑅 “ {𝐵}))
65ineq1d 3246 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) = ((𝑅 “ {𝐵}) ∩ 𝐴))
7 imass2 4885 . . . . . . 7 ({𝐵} ⊆ 𝐴 → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
82, 7syl 14 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ (𝑅𝐴))
9 simpl 108 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅𝐴) ⊆ 𝐴)
108, 9sstrd 3077 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ 𝐴)
11 df-ss 3054 . . . . 5 ((𝑅 “ {𝐵}) ⊆ 𝐴 ↔ ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
1210, 11sylib 121 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
136, 12eqtr2d 2151 . . 3 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴))
14 imainrect 4954 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴)
1513, 14syl6eqr 2168 . 2 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}))
16 df-ec 6399 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
17 df-ec 6399 . 2 [𝐵](𝑅 ∩ (𝐴 × 𝐴)) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵})
1815, 16, 173eqtr4g 2175 1 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  cin 3040  wss 3041  {csn 3497   × cxp 4507  cima 4512  [cec 6395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399
This theorem is referenced by:  qsinxp  6473  nqnq0pi  7214
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