Theorem resres 5983

Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)

Assertion

Ref	Expression
resres	⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))

Proof of Theorem resres

Step	Hyp	Ref	Expression
1		df-res 5663	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V))
2		df-res 5663	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
3	2	ineq1i 4162	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4		xpindir 5811	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
5	4	ineq2i 4163	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6		df-res 5663	. . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
7		inass 4173	. . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
8	5, 6, 7	3eqtr4ri 2795	. 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶))
9	1, 3, 8	3eqtri 2788	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3451   ∩ cin 3898   × cxp 5649   ↾ cres 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5657  df-rel 5658  df-res 5663

This theorem is used by:  rescom  5993  resabs1  5997  resmpt3  6030  resima2  6057  resdisj  6161  rescnvcnv  6204  fresin  6749  resdif  6844  curry1  8113  curry2  8116  frrlem4  8300  pmresg  8891  gruima  10880  rlimres  15718  lo1res  15719  rlimresb  15725  lo1eq  15728  rlimeq  15729  fsets  17340  setsid  17378  sscres  17991  gsumzres  20116  txkgen  23964  tsmsres  24456  ressxms  24837  ressms  24838  dvres  26224  dvres3a  26227  cpnres  26250  dvmptres3  26269  rlimcnp2  27287  df1stres  33290  df2ndres  33291  indf1ofs  33426  dfrcl2  44659  relexpaddss  44703  limsupresuz  46682  liminfresuz  46763  fouriersw  47210  fouriercn  47211  tposresg  49955  tposres3  49958