Theorem rescom 5906

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 4131	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
2	1	reseq2i 5877	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3		resres 5893	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))
4		resres 5893	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
5	2, 3, 4	3eqtr4i 2776	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1539   ∩ cin 3882   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-xp 5586  df-rel 5587  df-res 5592

This theorem is referenced by:  resabs2  5912  setscom  16809  dvres3a  24983  cpnres  25006  dvmptres3  25025  limsupresuz  43134  liminfresuz  43215