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Theorem resindm 5442
Description: Class restriction distributes over intersection. (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 5439 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
21ineq2d 3812 . 2 (Rel 𝐴 → ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴𝐵) ∩ 𝐴))
3 resindi 5410 . 2 (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴𝐵) ∩ (𝐴 ↾ dom 𝐴))
4 incom 3803 . . 3 ((𝐴𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴𝐵))
5 inres 5412 . . 3 (𝐴 ∩ (𝐴𝐵)) = ((𝐴𝐴) ↾ 𝐵)
6 inidm 3820 . . . 4 (𝐴𝐴) = 𝐴
76reseq1i 5390 . . 3 ((𝐴𝐴) ↾ 𝐵) = (𝐴𝐵)
84, 5, 73eqtrri 2648 . 2 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐴)
92, 3, 83eqtr4g 2680 1 (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  cin 3571  dom cdm 5112  cres 5114  Rel wrel 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-dm 5122  df-res 5124
This theorem is referenced by:  resdmdfsn  5443  resfnfinfin  8243  resfifsupp  8300  poimirlem3  33392  fresin2  39174  limsupvaluz  39746  cncfuni  39868  fourierdlem48  40140  fourierdlem49  40141  fourierdlem113  40205  sssmf  40716
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