Theorem xpss12 4833

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 3221	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3221	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 602	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4373	. 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 4731	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4731	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3270	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   C_ wss 3200   {copab 4149   X. cxp 4723

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-opab 4151  df-xp 4731

This theorem is referenced by:  xpss  4834  xpss1  4836  xpss2  4837  djussxp  4875  ssxpbm  5172  ssrnres  5179  cossxp  5259  cossxp2  5260  cocnvss  5262  relrelss  5263  fssxp  5502  oprabss  6106  pmss12g  6843  caserel  7285  casef  7286  dmaddpi  7544  dmmulpi  7545  rexpssxrxp  8223  ltrelxr  8239  dfz2  9551  phimullem  12796  znleval  14666  txuni2  14979  txbas  14981  neitx  14991  txcnp  14994  cnmpt2res  15020  psmetres2  15056  xmetres2  15102  metres2  15104  xmetresbl  15163  xmettx  15233  qtopbasss  15244  tgqioo  15278  resubmet  15279  limccnp2lem  15399  limccnp2cntop  15400  mpodvdsmulf1o  15713  fsumdvdsmul  15714