Theorem resindi 5954

Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Assertion

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

Proof of Theorem resindi

Step	Hyp	Ref	Expression
1		xpindir 5779	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
2	1	ineq2i 4149	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
3		inindi 4166	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
4	2, 3	eqtri 2764	. 2 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
5		df-res 5633	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
6		df-res 5633	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5633	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	ineq12i 4150	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
9	4, 5, 8	3eqtr4i 2774	1 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1548  Vcvv 3433   ∩ cin 3884   × cxp 5619   ↾ cres 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5138  df-xp 5627  df-rel 5628  df-res 5633

This theorem is referenced by:  resindm  5989  gsum2dlem2  19941  fnresin  32720  fressupp  32784  disjresdisj  38626