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Theorem xrnres2 34503
 Description: Two ways to express restriction of range Cartesian product, cf. xrnres 34502, xrnres3 34504. (Contributed by Peter Mazsa, 6-Sep-2021.)
Assertion
Ref Expression
xrnres2 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))

Proof of Theorem xrnres2
StepHypRef Expression
1 resco 5783 . . 3 (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = ((2nd ↾ (V × V)) ∘ (𝑆𝐴))
21ineq2i 3962 . 2 (((1st ↾ (V × V)) ∘ 𝑅) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
3 df-xrn 34475 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
43reseq1i 5530 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
5 inres 5555 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
64, 5eqtr4i 2796 . 2 ((𝑅𝑆) ↾ 𝐴) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
7 df-xrn 34475 . 2 (𝑅 ⋉ (𝑆𝐴)) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
82, 6, 73eqtr4i 2803 1 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631  Vcvv 3351   ∩ cin 3722   × cxp 5247  ◡ccnv 5248   ↾ cres 5251   ∘ ccom 5253  1st c1st 7313  2nd c2nd 7314   ⋉ 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 4915  ax-nul 4923  ax-pr 5034 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 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-co 5258  df-res 5261  df-xrn 34475 This theorem is referenced by:  xrnresex  34506  br1cossxrnres  34540
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