Theorem ssrin 3398

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 3187	. . . 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 3356	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3356	. . 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 3199	1 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   i^i cin 3165   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179

This theorem is referenced by:  sslin  3399  ssrind  3400  ss2in  3401  ssdisj  3517  ssdifin0  3542  ssres  4985  phplem2  6950  sbthlem7  7065  fiss  7079  tgss  14535  metrest  14978