Theorem ssrin 3432

Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ssrin	|- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )

Proof of Theorem ssrin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3221	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 336	. . 3 |- ( A C_ B -> ( ( x e. A /\ x e. C ) -> ( x e. B /\ x e. C ) ) )
3		elin 3390	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3390	. . 3 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
5	2, 3, 4	3imtr4g 205	. 2 |- ( A C_ B -> ( x e. ( A i^i C ) -> x e. ( B i^i C ) ) )
6	5	ssrdv 3233	1 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   i^i cin 3199   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  sslin  3433  ssrind  3434  ss2in  3435  ssdisj  3551  ssdifin0  3576  ssres  5039  phplem2  7038  sbthlem7  7161  fiss  7175  tgss  14786  metrest  15229