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Theorem resdifcom 5665
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 4098 . 2 ((𝐴𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2 df-res 5367 . 2 ((𝐴𝐶) ↾ 𝐵) = ((𝐴𝐶) ∩ (𝐵 × V))
3 df-res 5367 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
43difeq1i 3947 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
51, 2, 43eqtr4ri 2813 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  Vcvv 3398  cdif 3789  cin 3791   × cxp 5353  cres 5357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-in 3799  df-res 5367
This theorem is referenced by:  setsfun0  16291
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