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Theorem resdifdi 6187
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 5644 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
2 difxp1 6116 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
32ineq2i 4168 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4 indifdi 4242 . . 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 5644 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 5644 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7difeq12i 4079 . 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 1541  Vcvv 3444  cdif 3906  cin 3908   × cxp 5630  cres 5634
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-res 5644
This theorem is referenced by: (None)
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