Theorem rescom 4968

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 3352	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
2	1	reseq2i 4940	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ (𝐶 ∩ 𝐵))
3		resres 4955	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))
4		resres 4955	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
5	2, 3, 4	3eqtr4i 2224	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1364   ∩ cin 3153   ↾ cres 4662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-opab 4092  df-xp 4666  df-rel 4667  df-res 4672

This theorem is referenced by:  resabs2  4974  setscom  12661