Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrnres Structured version   Visualization version   GIF version

Theorem xrnres 38403
Description: Two ways to express restriction of range Cartesian product, see also xrnres2 38404, xrnres3 38405. (Contributed by Peter Mazsa, 5-Jun-2021.)
Assertion
Ref Expression
xrnres ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)

Proof of Theorem xrnres
StepHypRef Expression
1 resco 6270 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
21ineq1i 4216 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
3 df-xrn 38372 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
43reseq1i 5993 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5 inres2 38247 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
64, 5eqtr4i 2768 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
7 df-xrn 38372 . 2 ((𝑅𝐴) ⋉ 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
82, 6, 73eqtr4i 2775 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cin 3950   × cxp 5683  ccnv 5684  cres 5687  ccom 5689  1st c1st 8012  2nd c2nd 8013  cxrn 38181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-co 5694  df-res 5697  df-xrn 38372
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator