ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resdifcom GIF version

Theorem resdifcom 4837
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 3321 . 2 ((𝐴𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2 df-res 4551 . 2 ((𝐴𝐶) ↾ 𝐵) = ((𝐴𝐶) ∩ (𝐵 × V))
3 df-res 4551 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
43difeq1i 3190 . 2 ((𝐴𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
51, 2, 43eqtr4ri 2171 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1331  Vcvv 2686  cdif 3068  cin 3070   × cxp 4537  cres 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-res 4551
This theorem is referenced by:  setsfun0  12005
  Copyright terms: Public domain W3C validator