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Theorem resdifcom 5565
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 4006 . 2 ((𝐴𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2 df-res 5270 . 2 ((𝐴𝐶) ↾ 𝐵) = ((𝐴𝐶) ∩ (𝐵 × V))
3 df-res 5270 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
43difeq1i 3859 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
51, 2, 43eqtr4ri 2785 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1624  Vcvv 3332  cdif 3704  cin 3706   × cxp 5256  cres 5260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rab 3051  df-v 3334  df-dif 3710  df-in 3714  df-res 5270
This theorem is referenced by:  setsfun0  16088
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