Theorem xpss12 4884

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 3242	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2		ssel 3242	. . . 4 |- ( C C_ D -> ( y e. C -> y e. D ) )
3	1, 2	im2anan9 606	. . 3 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. A /\ y e. C ) -> ( x e. B /\ y e. D ) ) )
4	3	ssopab2dv 4423	. 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 4782	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
6		df-xp 4782	. 2 |- ( B X. D ) = { <. x , y >. | ( x e. B /\ y e. D ) }
7	4, 5, 6	3sstr4g 3291	1 |- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   C_ wss 3220   {copab 4192   X. cxp 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-opab 4194  df-xp 4782

This theorem is used by:  xpss  4885  xpss1  4887  xpss2  4888  djussxp  4927  ssxpbm  5226  ssrnres  5233  cossxp  5313  cossxp2  5314  cocnvss  5316  relrelss  5317  fssxp  5558  oprabss  6177  pmss12g  6960  caserel  7433  casef  7434  dmaddpi  7698  dmmulpi  7699  rexpssxrxp  8376  ltrelxr  8392  dfz2  9728  phimullem  13032  znleval  15097  txuni2  15482  txbas  15484  neitx  15494  txcnp  15497  cnmpt2res  15523  psmetres2  15559  xmetres2  15605  metres2  15607  xmetresbl  15666  xmettx  15736  qtopbasss  15747  tgqioo  15781  resubmet  15782  limccnp2lem  15902  limccnp2cntop  15903  mpodvdsmulf1o  16284  fsumdvdsmul  16285