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Theorem resdifdi 6190
Description: Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
Assertion
Ref Expression
resdifdi (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))

Proof of Theorem resdifdi
StepHypRef Expression
1 df-res 5632 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
2 difxp1 6119 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
32ineq2i 4148 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4 indifdi 4224 . . 3 (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
51, 3, 43eqtri 2768 . 2 (𝐴 ↾ (𝐵𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
6 df-res 5632 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 5632 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7difeq12i 4057 . 2 ((𝐴𝐵) ∖ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
95, 8eqtr4i 2767 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1548  Vcvv 3433  cdif 3881  cin 3883   × cxp 5618  cres 5622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364
This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5137  df-xp 5626  df-rel 5627  df-res 5632
This theorem is referenced by: (None)
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