Theorem rnxpss 5175

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 4742	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5162	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4938	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 5174	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3260	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3260	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3201   X. cxp 4729   `'ccnv 4730   dom cdm 4731   ran crn 4732

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  rnxpss2  5177  rnxpid  5178  ssxpbm  5179  ssxp2  5181  ssrnres  5186  funssxp  5512  fconst  5541  dff2  5799  fliftf  5950  tfrcllembfn  6566  frecuzrdgtcl  10720  cnconst2  15027  lmss  15040  exmidsbthrlem  16733