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Theorem metreslem 13751
Description: Lemma for metres 13754. (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 5119 . 2 (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ (𝑅 × 𝑅))
2 ineq2 3330 . . . 4 (dom 𝐷 = (𝑋 × 𝑋) → ((𝑅 × 𝑅) ∩ dom 𝐷) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋)))
3 dmres 4927 . . . 4 dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ dom 𝐷)
4 inxp 4760 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅))
5 incom 3327 . . . . 5 ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
64, 5eqtr3i 2200 . . . 4 ((𝑋𝑅) × (𝑋𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
72, 3, 63eqtr4g 2235 . . 3 (dom 𝐷 = (𝑋 × 𝑋) → dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑋𝑅) × (𝑋𝑅)))
87reseq2d 4906 . 2 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
91, 8eqtr3id 2224 1 (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋𝑅) × (𝑋𝑅))))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  cin 3128   × cxp 4623  dom cdm 4625  cres 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637
This theorem is referenced by:  xmetres  13753  metres  13754
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