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Theorem resdifcom 5842
 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 4176 . 2 ((𝐴𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2 df-res 5536 . 2 ((𝐴𝐶) ↾ 𝐵) = ((𝐴𝐶) ∩ (𝐵 × V))
3 df-res 5536 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
43difeq1i 4024 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
51, 2, 43eqtr4ri 2792 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  Vcvv 3409   ∖ cdif 3855   ∩ cin 3857   × cxp 5522   ↾ cres 5526 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-dif 3861  df-in 3865  df-res 5536 This theorem is referenced by:  setsfun0  16577  cycpmrn  30936  tocyccntz  30937
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