Theorem rnxpss 5214

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 4780	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5201	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4977	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 5213	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3280	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3280	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3220   X. cxp 4767   `'ccnv 4768   dom cdm 4769   ran crn 4770

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 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-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  rnxpss2  5216  rnxpid  5217  ssxpbm  5218  ssxp2  5220  ssrnres  5225  funssxp  5552  fconst  5583  dff2  5843  fliftf  5995  tfrcllembfn  6618  frecuzrdgtcl  10827  cnconst2  15257  lmss  15270  exmidsbthrlem  16972