Theorem ssrin 3265

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 3055	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 332	. . 3 |- ( A C_ B -> ( ( x e. A /\ x e. C ) -> ( x e. B /\ x e. C ) ) )
3		elin 3223	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3223	. . 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 3067	1 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1461   i^i cin 3034   C_ wss 3035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-ss 3048

This theorem is referenced by:  sslin  3266  ssrind  3267  ss2in  3268  ssdisj  3383  ssdifin0  3408  ssres  4801  phplem2  6698  sbthlem7  6801  fiss  6815  tgss  12069  metrest  12489