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Theorem metreslem 12581
Description: Lemma for metres 12584. (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 5036 . 2 (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ (𝑅 × 𝑅))
2 ineq2 3274 . . . 4 (dom 𝐷 = (𝑋 × 𝑋) → ((𝑅 × 𝑅) ∩ dom 𝐷) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋)))
3 dmres 4846 . . . 4 dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ dom 𝐷)
4 inxp 4679 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅))
5 incom 3271 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
64, 5eqtr3i 2163 . . . 4 ((𝑋𝑅) × (𝑋𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
72, 3, 63eqtr4g 2198 . . 3 (dom 𝐷 = (𝑋 × 𝑋) → dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅)))
87reseq2d 4825 . 2 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
91, 8syl5eqr 2187 1 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  cin 3073   × cxp 4543  dom cdm 4545  cres 4547
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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-xp 4551  df-rel 4552  df-cnv 4553  df-dm 4555  df-rn 4556  df-res 4557
This theorem is referenced by:  xmetres  12583  metres  12584
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