Theorem resres 5947

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 5635	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V))
2		df-res 5635	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
3	2	ineq1i 4169	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4		xpindir 5781	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
5	4	ineq2i 4170	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6		df-res 5635	. . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
7		inass 4181	. . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
8	5, 6, 7	3eqtr4ri 2763	. 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶))
9	1, 3, 8	3eqtri 2756	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1540  Vcvv 3438   ∩ cin 3904   × cxp 5621   ↾ cres 5625

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 2701  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 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5158  df-xp 5629  df-rel 5630  df-res 5635

This theorem is referenced by:  rescom  5957  resabs1  5961  resima2  5971  resmpt3  5993  resdisj  6122  rescnvcnv  6157  fresin  6697  resdif  6789  curry1  8044  curry2  8047  frrlem4  8229  pmresg  8804  gruima  10715  rlimres  15484  lo1res  15485  rlimresb  15491  lo1eq  15494  rlimeq  15495  fsets  17099  setsid  17137  sscres  17749  gsumzres  19807  txkgen  23556  tsmsres  24048  ressxms  24430  ressms  24431  dvres  25829  dvres3a  25832  cpnres  25856  dvmptres3  25877  rlimcnp2  26893  df1stres  32665  df2ndres  32666  indf1ofs  32828  dfrcl2  43667  relexpaddss  43711  limsupresuz  45704  liminfresuz  45785  fouriersw  46232  fouriercn  46233  tposresg  48882  tposres3  48885