Theorem rnxpss 5194

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 4760	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5181	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4957	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 5193	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3270	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3270	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3211   X. cxp 4747   `'ccnv 4748   dom cdm 4749   ran crn 4750

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-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  rnxpss2  5196  rnxpid  5197  ssxpbm  5198  ssxp2  5200  ssrnres  5205  funssxp  5532  fconst  5563  dff2  5821  fliftf  5972  tfrcllembfn  6588  frecuzrdgtcl  10774  cnconst2  15098  lmss  15111  exmidsbthrlem  16802