Theorem ssrin 3223

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 3017	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 329	. . 3 |- ( A C_ B -> ( ( x e. A /\ x e. C ) -> ( x e. B /\ x e. C ) ) )
3		elin 3181	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3181	. . 3 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
5	2, 3, 4	3imtr4g 203	. 2 |- ( A C_ B -> ( x e. ( A i^i C ) -> x e. ( B i^i C ) ) )
6	5	ssrdv 3029	1 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1438   i^i cin 2996   C_ wss 2997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010

This theorem is referenced by:  sslin  3224  ss2in  3225  ssdisj  3336  ssdifin0  3360  ssres  4726  phplem2  6549  sbthlem7  6651