Theorem resindi 5019

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 4857	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
2	1	ineq2i 3402	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
3		inindi 3421	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
4	2, 3	eqtri 2250	. 2 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
5		df-res 4730	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
6		df-res 4730	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 4730	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	ineq12i 3403	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
9	4, 5, 8	3eqtr4i 2260	1 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1395  Vcvv 2799   ∩ cin 3196   × cxp 4716   ↾ cres 4720

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-xp 4724  df-rel 4725  df-res 4730

This theorem is referenced by:  resindm  5046