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Theorem xrnres 35652
Description: Two ways to express restriction of range Cartesian product, see also xrnres2 35653, xrnres3 35654. (Contributed by Peter Mazsa, 5-Jun-2021.)
Assertion
Ref Expression
xrnres ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)

Proof of Theorem xrnres
StepHypRef Expression
1 resco 6105 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
21ineq1i 4187 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
3 df-xrn 35625 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
43reseq1i 5851 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5 inres2 35508 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
64, 5eqtr4i 2849 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
7 df-xrn 35625 . 2 ((𝑅𝐴) ⋉ 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
82, 6, 73eqtr4i 2856 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3496  cin 3937   × cxp 5555  ccnv 5556  cres 5559  ccom 5561  1st c1st 7689  2nd c2nd 7690  cxrn 35454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-co 5566  df-res 5569  df-xrn 35625
This theorem is referenced by: (None)
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