MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metreslem Structured version   Visualization version   GIF version

Theorem metreslem 24223
Description: Lemma for metres 24226. (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 6225 . 2 (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ (𝑅 × 𝑅))
2 ineq2 4201 . . . 4 (dom 𝐷 = (𝑋 × 𝑋) → ((𝑅 × 𝑅) ∩ dom 𝐷) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋)))
3 dmres 5997 . . . 4 dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ dom 𝐷)
4 inxp 5825 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅))
5 incom 4196 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
64, 5eqtr3i 2756 . . . 4 ((𝑋𝑅) × (𝑋𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
72, 3, 63eqtr4g 2791 . . 3 (dom 𝐷 = (𝑋 × 𝑋) → dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅)))
87reseq2d 5975 . 2 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
91, 8eqtr3id 2780 1 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3942   × cxp 5667  dom cdm 5669  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681
This theorem is referenced by:  xmetres  24225  metres  24226
  Copyright terms: Public domain W3C validator