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Theorem resdifdi 6192
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 5649 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
2 difxp1 6121 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
32ineq2i 4173 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4 indifdi 4247 . . 3 (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
51, 3, 43eqtri 2765 . 2 (𝐴 ↾ (𝐵𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
6 df-res 5649 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 5649 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7difeq12i 4084 . 2 ((𝐴𝐵) ∖ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
95, 8eqtr4i 2764 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-res 5649
This theorem is referenced by: (None)
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