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  6107  pmss12g  6844  caserel  7286  casef  7287  dmaddpi  7545  dmmulpi  7546  rexpssxrxp  8224  ltrelxr  8240  dfz2  9552  phimullem  12815  znleval  14686  txuni2  14999  txbas  15001  neitx  15011  txcnp  15014  cnmpt2res  15040  psmetres2  15076  xmetres2  15122  metres2  15124  xmetresbl  15183  xmettx  15253  qtopbasss  15264  tgqioo  15298  resubmet  15299  limccnp2lem  15419  limccnp2cntop  15420  mpodvdsmulf1o  15733  fsumdvdsmul  15734