Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  resisresindm Structured version   Visualization version   GIF version

Theorem resisresindm 40320
 Description: The restriction of a relation by a set 𝐵 is identical with the restriction by the intersection of 𝐵 with the domain of the relation. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
Assertion
Ref Expression
resisresindm (Rel 𝐹 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))

Proof of Theorem resisresindm
StepHypRef Expression
1 resdm 5361 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
21eqcomd 2616 . . 3 (Rel 𝐹𝐹 = (𝐹 ↾ dom 𝐹))
32ineq2d 3776 . 2 (Rel 𝐹 → ((𝐹𝐵) ∩ 𝐹) = ((𝐹𝐵) ∩ (𝐹 ↾ dom 𝐹)))
4 resss 5342 . . . 4 (𝐹𝐵) ⊆ 𝐹
5 df-ss 3554 . . . 4 ((𝐹𝐵) ⊆ 𝐹 ↔ ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵))
64, 5mpbi 219 . . 3 ((𝐹𝐵) ∩ 𝐹) = (𝐹𝐵)
76eqcomi 2619 . 2 (𝐹𝐵) = ((𝐹𝐵) ∩ 𝐹)
8 resindi 5332 . 2 (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = ((𝐹𝐵) ∩ (𝐹 ↾ dom 𝐹))
93, 7, 83eqtr4g 2669 1 (Rel 𝐹 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050 This theorem is referenced by:  resfnfinfin  40339
 Copyright terms: Public domain W3C validator