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Theorem resdifdi 6223
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 5659 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
2 difxp1 6150 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
32ineq2i 4169 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4 indifdi 4246 . . 3 (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
51, 3, 43eqtri 2789 . 2 (𝐴 ↾ (𝐵𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
6 df-res 5659 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 5659 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7difeq12i 4078 . 2 ((𝐴𝐵) ∖ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
95, 8eqtr4i 2788 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1560  Vcvv 3454  cdif 3901  cin 3903   × cxp 5645  cres 5649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5653  df-rel 5654  df-res 5659
This theorem is referenced by: (None)
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