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Theorem ecinxp 6544
 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 3697 . . . . . . 7 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
3 df-ss 3111 . . . . . . 7 ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
42, 3sylib 121 . . . . . 6 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ({𝐵} ∩ 𝐴) = {𝐵})
54imaeq2d 4921 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ ({𝐵} ∩ 𝐴)) = (𝑅 “ {𝐵}))
65ineq1d 3303 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) = ((𝑅 “ {𝐵}) ∩ 𝐴))
7 imass2 4955 . . . . . . 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 3134 . . . . 5 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) ⊆ 𝐴)
11 df-ss 3111 . . . . 5 ((𝑅 “ {𝐵}) ⊆ 𝐴 ↔ ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
1210, 11sylib 121 . . . 4 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
136, 12eqtr2d 2188 . . 3 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴))
14 imainrect 5024 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴)
1513, 14eqtr4di 2205 . 2 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → (𝑅 “ {𝐵}) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}))
16 df-ec 6471 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
17 df-ec 6471 . 2 [𝐵](𝑅 ∩ (𝐴 × 𝐴)) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵})
1815, 16, 173eqtr4g 2212 1 (((𝑅𝐴) ⊆ 𝐴𝐵𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125   ∩ cin 3097   ⊆ wss 3098  {csn 3556   × cxp 4577   “ cima 4582  [cec 6467 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-xp 4585  df-rel 4586  df-cnv 4587  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-ec 6471 This theorem is referenced by:  qsinxp  6545  nqnq0pi  7337
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