Theorem xpss12 4770

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 3177	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3177	. . . 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 4313	. 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 4669	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4669	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3226	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   C_ wss 3157   {copab 4093   X. cxp 4661

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-opab 4095  df-xp 4669

This theorem is referenced by:  xpss  4771  xpss1  4773  xpss2  4774  djussxp  4811  ssxpbm  5105  ssrnres  5112  cossxp  5192  cossxp2  5193  cocnvss  5195  relrelss  5196  fssxp  5425  oprabss  6008  pmss12g  6734  caserel  7153  casef  7154  dmaddpi  7392  dmmulpi  7393  rexpssxrxp  8071  ltrelxr  8087  dfz2  9398  phimullem  12393  znleval  14209  txuni2  14492  txbas  14494  neitx  14504  txcnp  14507  cnmpt2res  14533  psmetres2  14569  xmetres2  14615  metres2  14617  xmetresbl  14676  xmettx  14746  qtopbasss  14757  tgqioo  14791  resubmet  14792  limccnp2lem  14912  limccnp2cntop  14913  mpodvdsmulf1o  15226  fsumdvdsmul  15227