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

Theorem resindm 5602
 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)) = (𝐴𝐵))

Proof of Theorem resindm
StepHypRef Expression
1 resdm 5599 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
21ineq2d 3957 . 2 (Rel 𝐴 → ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴𝐵) ∩ 𝐴))
3 resindi 5570 . 2 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴))
4 incom 3948 . . 3 ((𝐴𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴𝐵))
5 inres 5572 . . 3 (𝐴 ∩ (𝐴𝐵)) = ((𝐴𝐴) ↾ 𝐵)
6 inidm 3965 . . . 4 (𝐴𝐴) = 𝐴
76reseq1i 5547 . . 3 ((𝐴𝐴) ↾ 𝐵) = (𝐴𝐵)
84, 5, 73eqtrri 2787 . 2 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐴)
92, 3, 83eqtr4g 2819 1 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∩ cin 3714  dom cdm 5266   ↾ cres 5268  Rel wrel 5271 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-dm 5276  df-res 5278 This theorem is referenced by:  resdmdfsn  5603  resfnfinfin  8413  resfifsupp  8470  poimirlem3  33743  fresin2  39869  limsupvaluz  40461  cncfuni  40620  fourierdlem48  40892  fourierdlem49  40893  fourierdlem113  40957  sssmf  41471
 Copyright terms: Public domain W3C validator