Theorem xpss12 4862

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 3236	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3236	. . . 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 4402	. 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 4760	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4760	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3285	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   C_ wss 3214   {copab 4175   X. cxp 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3220  df-ss 3227  df-opab 4177  df-xp 4760

This theorem is referenced by:  xpss  4863  xpss1  4865  xpss2  4866  djussxp  4905  ssxpbm  5203  ssrnres  5210  cossxp  5290  cossxp2  5291  cocnvss  5293  relrelss  5294  fssxp  5535  oprabss  6147  pmss12g  6922  caserel  7391  casef  7392  dmaddpi  7656  dmmulpi  7657  rexpssxrxp  8334  ltrelxr  8350  dfz2  9667  phimullem  12947  znleval  14927  txuni2  15247  txbas  15249  neitx  15259  txcnp  15262  cnmpt2res  15288  psmetres2  15324  xmetres2  15370  metres2  15372  xmetresbl  15431  xmettx  15501  qtopbasss  15512  tgqioo  15546  resubmet  15547  limccnp2lem  15667  limccnp2cntop  15668  mpodvdsmulf1o  15984  fsumdvdsmul  15985