Theorem rnxpss 4926

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 4508	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 4913	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4698	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 4925	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3093	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3093	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3035   X. cxp 4495   `'ccnv 4496   dom cdm 4497   ran crn 4498

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-cnv 4505  df-dm 4507  df-rn 4508

This theorem is referenced by:  rnxpss2  4928  rnxpid  4929  ssxpbm  4930  ssxp2  4932  ssrnres  4937  funssxp  5248  fconst  5274  dff2  5516  fliftf  5652  tfrcllembfn  6206  frecuzrdgtcl  10072  cnconst2  12238  lmss  12251  exmidsbthrlem  12898