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Theorem xrnres3 34591
Description: Two ways to express restriction of range Cartesian product, cf. xrnres 34589, xrnres2 34590. (Contributed by Peter Mazsa, 28-Mar-2020.)
Assertion
Ref Expression
xrnres3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Proof of Theorem xrnres3
StepHypRef Expression
1 resco 5825 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
2 resco 5825 . . 3 (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = ((2nd ↾ (V × V)) ∘ (𝑆𝐴))
31, 2ineq12i 3974 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
4 df-xrn 34562 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
54reseq1i 5561 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6 resindir 5589 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
75, 6eqtri 2787 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
8 df-xrn 34562 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
93, 7, 83eqtr4i 2797 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  Vcvv 3350  cin 3731   × cxp 5275  ccnv 5276  cres 5279  ccom 5281  1st c1st 7364  2nd c2nd 7365  cxrn 34403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-co 5286  df-res 5289  df-xrn 34562
This theorem is referenced by:  xrnres4  34592  xrnresex  34593
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