Theorem resres 5948

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 5633	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V))
2		df-res 5633	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
3	2	ineq1i 4165	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4		xpindir 5780	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
5	4	ineq2i 4166	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6		df-res 5633	. . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
7		inass 4177	. . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
8	5, 6, 7	3eqtr4ri 2767	. 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶))
9	1, 3, 8	3eqtri 2760	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1541  Vcvv 3437   ∩ cin 3897   × cxp 5619   ↾ cres 5623

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5158  df-xp 5627  df-rel 5628  df-res 5633

This theorem is referenced by:  rescom  5958  resabs1  5962  resima2  5972  resmpt3  5994  resdisj  6124  rescnvcnv  6159  fresin  6700  resdif  6792  curry1  8043  curry2  8046  frrlem4  8228  pmresg  8804  gruima  10704  rlimres  15472  lo1res  15473  rlimresb  15479  lo1eq  15482  rlimeq  15483  fsets  17087  setsid  17125  sscres  17738  gsumzres  19829  txkgen  23587  tsmsres  24079  ressxms  24460  ressms  24461  dvres  25859  dvres3a  25862  cpnres  25886  dvmptres3  25907  rlimcnp2  26923  df1stres  32709  df2ndres  32710  indf1ofs  32876  dfrcl2  43831  relexpaddss  43875  limsupresuz  45863  liminfresuz  45944  fouriersw  46391  fouriercn  46392  tposresg  49039  tposres3  49042