Theorem rnxpss 5159

Description: The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
rnxpss	|- ran ( A X. B ) C_ B

Proof of Theorem rnxpss

Step	Hyp	Ref	Expression
1		df-rn 4729	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5146	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4923	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 5158	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3256	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3256	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3197   X. cxp 4716   `'ccnv 4717   dom cdm 4718   ran crn 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  rnxpss2  5161  rnxpid  5162  ssxpbm  5163  ssxp2  5165  ssrnres  5170  funssxp  5492  fconst  5520  dff2  5778  fliftf  5922  tfrcllembfn  6501  frecuzrdgtcl  10629  cnconst2  14901  lmss  14914  exmidsbthrlem  16349