Theorem resindm 6050

Description: When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)

Assertion

Ref	Expression
resindm	⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))

Proof of Theorem resindm

Step	Hyp	Ref	Expression
1		resdm 6046	. . 3 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
2	1	ineq2d 4228	. 2 ⊢ (Rel 𝐴 → ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ 𝐴))
3		resindi 6016	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴))
4		incom 4217	. . 3 ⊢ ((𝐴 ↾ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴 ↾ 𝐵))
5		inres 6018	. . 3 ⊢ (𝐴 ∩ (𝐴 ↾ 𝐵)) = ((𝐴 ∩ 𝐴) ↾ 𝐵)
6		inidm 4235	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	6	reseq1i 5996	. . 3 ⊢ ((𝐴 ∩ 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵)
8	4, 5, 7	3eqtrri 2768	. 2 ⊢ (𝐴 ↾ 𝐵) = ((𝐴 ↾ 𝐵) ∩ 𝐴)
9	2, 3, 8	3eqtr4g 2800	1 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3962  dom cdm 5689   ↾ cres 5691  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701

This theorem is referenced by:  resdmdfsn  6051  resfnfinfin  9375  resfifsupp  9435  poimirlem3  37610  fresin2  45115  limsupvaluz  45664  cncfuni  45842  fourierdlem48  46110  fourierdlem49  46111  fourierdlem113  46175  sssmf  46694