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Theorem xrnres3 35532
Description: Two ways to express restriction of range Cartesian product, see also xrnres 35530, xrnres2 35531. (Contributed by Peter Mazsa, 28-Mar-2020.)
Assertion
Ref Expression
xrnres3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Proof of Theorem xrnres3
StepHypRef Expression
1 resco 6096 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
2 resco 6096 . . 3 (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = ((2nd ↾ (V × V)) ∘ (𝑆𝐴))
31, 2ineq12i 4184 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
4 df-xrn 35503 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
54reseq1i 5842 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6 resindir 5863 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
75, 6eqtri 2841 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
8 df-xrn 35503 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
93, 7, 83eqtr4i 2851 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  Vcvv 3492  cin 3932   × cxp 5546  ccnv 5547  cres 5550  ccom 5552  1st c1st 7676  2nd c2nd 7677  cxrn 35333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-co 5557  df-res 5560  df-xrn 35503
This theorem is referenced by:  xrnres4  35533  xrnresex  35534
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