Theorem rescom 5975

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 4174	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
2	1	reseq2i 5949	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3		resres 5965	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))
4		resres 5965	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
5	2, 3, 4	3eqtr4i 2763	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1540   ∩ cin 3915   ↾ cres 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5172  df-xp 5646  df-rel 5647  df-res 5652

This theorem is referenced by:  resabs2  5982  setscom  17156  dvres3a  25821  cpnres  25845  dvmptres3  25866  limsupresuz  45694  liminfresuz  45775  tposresg  48856