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Theorem xrnres 38048
Description: Two ways to express restriction of range Cartesian product, see also xrnres2 38049, xrnres3 38050. (Contributed by Peter Mazsa, 5-Jun-2021.)
Assertion
Ref Expression
xrnres ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)

Proof of Theorem xrnres
StepHypRef Expression
1 resco 6260 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
21ineq1i 4208 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
3 df-xrn 38017 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
43reseq1i 5984 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5 inres2 37891 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
64, 5eqtr4i 2756 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
7 df-xrn 38017 . 2 ((𝑅𝐴) ⋉ 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
82, 6, 73eqtr4i 2763 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3461  cin 3945   × cxp 5679  ccnv 5680  cres 5683  ccom 5685  1st c1st 8000  2nd c2nd 8001  cxrn 37823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5687  df-rel 5688  df-co 5690  df-res 5693  df-xrn 38017
This theorem is referenced by: (None)
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