Theorem ssrin 3328

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 3118	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 334	. . 3 |- ( A C_ B -> ( ( x e. A /\ x e. C ) -> ( x e. B /\ x e. C ) ) )
3		elin 3286	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3286	. . 3 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
5	2, 3, 4	3imtr4g 204	. 2 |- ( A C_ B -> ( x e. ( A i^i C ) -> x e. ( B i^i C ) ) )
6	5	ssrdv 3130	1 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2125   i^i cin 3097   C_ wss 3098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-in 3104  df-ss 3111

This theorem is referenced by:  sslin  3329  ssrind  3330  ss2in  3331  ssdisj  3446  ssdifin0  3471  ssres  4885  phplem2  6787  sbthlem7  6896  fiss  6910  tgss  12410  metrest  12853