Theorem xpss12 4782

Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
xpss12	|- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Proof of Theorem xpss12

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3187	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3187	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 598	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4325	. 2 |- ( ( A C_ B /\ C C_ D ) -> { <. x , y >. | ( x e. A /\ y e. C ) } C_ { <. x , y >. | ( x e. B /\ y e. D ) } )
5		df-xp 4681	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4681	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3236	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   C_ wss 3166   {copab 4104   X. cxp 4673

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-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-opab 4106  df-xp 4681

This theorem is referenced by:  xpss  4783  xpss1  4785  xpss2  4786  djussxp  4823  ssxpbm  5118  ssrnres  5125  cossxp  5205  cossxp2  5206  cocnvss  5208  relrelss  5209  fssxp  5443  oprabss  6031  pmss12g  6762  caserel  7189  casef  7190  dmaddpi  7438  dmmulpi  7439  rexpssxrxp  8117  ltrelxr  8133  dfz2  9445  phimullem  12547  znleval  14415  txuni2  14728  txbas  14730  neitx  14740  txcnp  14743  cnmpt2res  14769  psmetres2  14805  xmetres2  14851  metres2  14853  xmetresbl  14912  xmettx  14982  qtopbasss  14993  tgqioo  15027  resubmet  15028  limccnp2lem  15148  limccnp2cntop  15149  mpodvdsmulf1o  15462  fsumdvdsmul  15463