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Theorem xrnres3 35724
Description: Two ways to express restriction of range Cartesian product, see also xrnres 35722, xrnres2 35723. (Contributed by Peter Mazsa, 28-Mar-2020.)
Assertion
Ref Expression
xrnres3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Proof of Theorem xrnres3
StepHypRef Expression
1 resco 6091 . . 3 (((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) = ((1st ↾ (V × V)) ∘ (𝑅𝐴))
2 resco 6091 . . 3 (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴) = ((2nd ↾ (V × V)) ∘ (𝑆𝐴))
31, 2ineq12i 4172 . 2 ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
4 df-xrn 35695 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
54reseq1i 5837 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴)
6 resindir 5858 . . 3 ((((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆)) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
75, 6eqtri 2847 . 2 ((𝑅𝑆) ↾ 𝐴) = ((((1st ↾ (V × V)) ∘ 𝑅) ↾ 𝐴) ∩ (((2nd ↾ (V × V)) ∘ 𝑆) ↾ 𝐴))
8 df-xrn 35695 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (((1st ↾ (V × V)) ∘ (𝑅𝐴)) ∩ ((2nd ↾ (V × V)) ∘ (𝑆𝐴)))
93, 7, 83eqtr4i 2857 1 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  Vcvv 3480  cin 3918   × cxp 5541  ccnv 5542  cres 5545  ccom 5547  1st c1st 7679  2nd c2nd 7680  cxrn 35524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-xp 5549  df-rel 5550  df-co 5552  df-res 5555  df-xrn 35695
This theorem is referenced by:  xrnres4  35725  xrnresex  35726
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