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Theorem rnxpss 5018
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 × 𝐵) ⊆ 𝐵

Proof of Theorem rnxpss
StepHypRef Expression
1 df-rn 4598 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5005 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 4788 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 5017 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3160 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3160 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff set class
Syntax hints:  wss 3102   × cxp 4585  ccnv 4586  dom cdm 4587  ran crn 4588
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-rel 4594  df-cnv 4595  df-dm 4597  df-rn 4598
This theorem is referenced by:  rnxpss2  5020  rnxpid  5021  ssxpbm  5022  ssxp2  5024  ssrnres  5029  funssxp  5340  fconst  5366  dff2  5612  fliftf  5750  tfrcllembfn  6305  frecuzrdgtcl  10315  cnconst2  12675  lmss  12688  exmidsbthrlem  13635
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