Theorem rnxpss 5029

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 4609	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5016	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4799	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 5028	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3169	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3169	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3111   X. cxp 4596   `'ccnv 4597   dom cdm 4598   ran crn 4599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-rel 4605  df-cnv 4606  df-dm 4608  df-rn 4609

This theorem is referenced by:  rnxpss2  5031  rnxpid  5032  ssxpbm  5033  ssxp2  5035  ssrnres  5040  funssxp  5351  fconst  5377  dff2  5623  fliftf  5761  tfrcllembfn  6316  frecuzrdgtcl  10337  cnconst2  12774  lmss  12787  exmidsbthrlem  13735