Theorem rescom 5946

Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)

Assertion

Ref	Expression
rescom	⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵)

Proof of Theorem rescom

Step	Hyp	Ref	Expression
1		incom 4154	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
2	1	reseq2i 5920	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3		resres 5936	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))
4		resres 5936	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
5	2, 3, 4	3eqtr4i 2764	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1541   ∩ cin 3896   ↾ cres 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-opab 5149  df-xp 5617  df-rel 5618  df-res 5623

This theorem is referenced by:  resabs2  5953  setscom  17086  dvres3a  25837  cpnres  25861  dvmptres3  25882  limsupresuz  45741  liminfresuz  45822  tposresg  48909