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Theorem resdifcom 6004
Description: Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
Assertion
Ref Expression
resdifcom ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)

Proof of Theorem resdifcom
StepHypRef Expression
1 indif1 4270 . 2 ((𝐴𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2 df-res 5690 . 2 ((𝐴𝐶) ↾ 𝐵) = ((𝐴𝐶) ∩ (𝐵 × V))
3 df-res 5690 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
43difeq1i 4114 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
51, 2, 43eqtr4ri 2764 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3461  cdif 3941  cin 3943   × cxp 5676  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-in 3951  df-res 5690
This theorem is referenced by:  setsfun0  17144  cycpmrn  32956  tocyccntz  32957
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