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Theorem xrnres 38384
Description: Two ways to express restriction of range Cartesian product, see also xrnres2 38385, xrnres3 38386. (Contributed by Peter Mazsa, 5-Jun-2021.)
Assertion
Ref Expression
xrnres ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)

Proof of Theorem xrnres
StepHypRef Expression
1 resco 6272 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
21ineq1i 4224 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
3 df-xrn 38353 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
43reseq1i 5996 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5 inres2 38227 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
64, 5eqtr4i 2766 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
7 df-xrn 38353 . 2 ((𝑅𝐴) ⋉ 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
82, 6, 73eqtr4i 2773 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cin 3962   × cxp 5687  ccnv 5688  cres 5691  ccom 5693  1st c1st 8011  2nd c2nd 8012  cxrn 38161
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-co 5698  df-res 5701  df-xrn 38353
This theorem is referenced by: (None)
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