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Theorem metreslem 23661
Description: Lemma for metres 23664. (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 6182 . 2 (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ (𝑅 × 𝑅))
2 ineq2 4164 . . . 4 (dom 𝐷 = (𝑋 × 𝑋) → ((𝑅 × 𝑅) ∩ dom 𝐷) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋)))
3 dmres 5957 . . . 4 dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ dom 𝐷)
4 inxp 5786 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅))
5 incom 4159 . . . . 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 5935 . 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 1541  cin 3907   × cxp 5629  dom cdm 5631  cres 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643
This theorem is referenced by:  xmetres  23663  metres  23664
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