Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrnres3 Structured version   Visualization version   GIF version

Theorem xrnres3 38104
Description: Two ways to express restriction of range Cartesian product, see also xrnres 38102, xrnres2 38103. (Contributed by Peter Mazsa, 28-Mar-2020.)
Assertion
Ref Expression
xrnres3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Proof of Theorem xrnres3
StepHypRef Expression
1 resco 6263 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
2 resco 6263 . . 3 (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = ((2nd ↾ (V × V)) ∘ (𝑆𝐴))
31, 2ineq12i 4211 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
4 df-xrn 38071 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
54reseq1i 5987 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6 resindir 6008 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
75, 6eqtri 2754 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
8 df-xrn 38071 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
93, 7, 83eqtr4i 2764 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  Vcvv 3462  cin 3946   × cxp 5682  ccnv 5683  cres 5686  ccom 5688  1st c1st 8003  2nd c2nd 8004  cxrn 37877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5156  df-opab 5218  df-xp 5690  df-rel 5691  df-co 5693  df-res 5696  df-xrn 38071
This theorem is referenced by:  xrnres4  38105  xrnresex  38106
  Copyright terms: Public domain W3C validator