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Theorem xrnres3 34504
Description: Two ways to express restriction of range Cartesian product, cf. xrnres 34502, xrnres2 34503. (Contributed by Peter Mazsa, 28-Mar-2020.)
Assertion
Ref Expression
xrnres3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Proof of Theorem xrnres3
StepHypRef Expression
1 resco 5782 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
2 resco 5782 . . 3 (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = ((2nd ↾ (V × V)) ∘ (𝑆𝐴))
31, 2ineq12i 3963 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
4 df-xrn 34475 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
54reseq1i 5529 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6 resindir 5553 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
75, 6eqtri 2793 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
8 df-xrn 34475 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
93, 7, 83eqtr4i 2803 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  Vcvv 3351  cin 3722   × cxp 5248  ccnv 5249  cres 5252  ccom 5254  1st c1st 7317  2nd c2nd 7318  cxrn 34314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-co 5259  df-res 5262  df-xrn 34475
This theorem is referenced by:  xrnres4  34505  xrnresex  34506
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