Theorem rnxpss 5123

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 4694	. 2 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5110	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4888	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4		dmxpss 5122	. . 3 |- dom ( B X. A ) C_ B
5	3, 4	eqsstri 3229	. 2 |- dom `' ( A X. B ) C_ B
6	1, 5	eqsstri 3229	1 |- ran ( A X. B ) C_ B

Syntax hints:   C_ wss 3170   X. cxp 4681   `'ccnv 4682   dom cdm 4683   ran crn 4684

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-rel 4690  df-cnv 4691  df-dm 4693  df-rn 4694

This theorem is referenced by:  rnxpss2  5125  rnxpid  5126  ssxpbm  5127  ssxp2  5129  ssrnres  5134  funssxp  5455  fconst  5483  dff2  5737  fliftf  5881  tfrcllembfn  6456  frecuzrdgtcl  10579  cnconst2  14780  lmss  14793  exmidsbthrlem  16102