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Theorem resdifdir 6193
Description: Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
Assertion
Ref Expression
resdifdir ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Proof of Theorem resdifdir
StepHypRef Expression
1 indifdir 4248 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
2 df-res 5649 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 5649 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 5649 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4difeq12i 4084 . 2 ((𝐴𝐶) ∖ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2771 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3447  cdif 3911  cin 3913   × cxp 5635  cres 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-in 3921  df-res 5649
This theorem is referenced by: (None)
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