Theorem metreslem 23260

Description: Lemma for metres 23263. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
metreslem	⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))))

Proof of Theorem metreslem

Step	Hyp	Ref	Expression
1		resdmres 6095	. 2 ⊢ (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ (𝑅 × 𝑅))
2		ineq2 4121	. . . 4 ⊢ (dom 𝐷 = (𝑋 × 𝑋) → ((𝑅 × 𝑅) ∩ dom 𝐷) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋)))
3		dmres 5873	. . . 4 ⊢ dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ dom 𝐷)
4		inxp 5701	. . . . 5 ⊢ ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))
5		incom 4115	. . . . 5 ⊢ ((𝑋 × 𝑋) ∩ (𝑅 × 𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
6	4, 5	eqtr3i 2767	. . . 4 ⊢ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅)) = ((𝑅 × 𝑅) ∩ (𝑋 × 𝑋))
7	2, 3, 6	3eqtr4g 2803	. . 3 ⊢ (dom 𝐷 = (𝑋 × 𝑋) → dom (𝐷 ↾ (𝑅 × 𝑅)) = ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅)))
8	7	reseq2d 5851	. 2 ⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ dom (𝐷 ↾ (𝑅 × 𝑅))) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))))
9	1, 8	eqtr3id 2792	1 ⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))))

Syntax hints:   → wi 4   = wceq 1543   ∩ cin 3865   × cxp 5549  dom cdm 5551   ↾ cres 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563

This theorem is referenced by:  xmetres  23262  metres  23263