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Theorem metreslem 24373
Description: Lemma for metres 24376. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
metreslem (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))

Proof of Theorem metreslem
StepHypRef Expression
1 resdmres 6251 . 2 (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ (𝑅 × 𝑅))
2 ineq2 4213 . . . 4 (dom 𝐷 = (𝑋 × 𝑋) → ((𝑅 × 𝑅) ∩ dom 𝐷) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋)))
3 dmres 6029 . . . 4 dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ dom 𝐷)
4 inxp 5841 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅))
5 incom 4208 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
64, 5eqtr3i 2766 . . . 4 ((𝑋𝑅) × (𝑋𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
72, 3, 63eqtr4g 2801 . . 3 (dom 𝐷 = (𝑋 × 𝑋) → dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅)))
87reseq2d 5996 . 2 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
91, 8eqtr3id 2790 1 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cin 3949   × cxp 5682  dom cdm 5684  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696
This theorem is referenced by:  xmetres  24375  metres  24376
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