Theorem rnxpss 4970

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 4550	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 4957	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4740	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 4969	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3129	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3129	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3071   X. cxp 4537   `'ccnv 4538   dom cdm 4539   ran crn 4540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550

This theorem is referenced by:  rnxpss2  4972  rnxpid  4973  ssxpbm  4974  ssxp2  4976  ssrnres  4981  funssxp  5292  fconst  5318  dff2  5564  fliftf  5700  tfrcllembfn  6254  frecuzrdgtcl  10185  cnconst2  12402  lmss  12415  exmidsbthrlem  13217