Theorem rnxpss 5168

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 4736	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5155	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4932	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 5167	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3259	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3259	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3200   X. cxp 4723   `'ccnv 4724   dom cdm 4725   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  rnxpss2  5170  rnxpid  5171  ssxpbm  5172  ssxp2  5174  ssrnres  5179  funssxp  5504  fconst  5532  dff2  5791  fliftf  5939  tfrcllembfn  6522  frecuzrdgtcl  10673  cnconst2  14956  lmss  14969  exmidsbthrlem  16626