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Theorem rnxpss 5078
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 4655 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5065 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 4846 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 5077 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3202 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3202 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff set class
Syntax hints:  wss 3144   × cxp 4642  ccnv 4643  dom cdm 4644  ran crn 4645
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655
This theorem is referenced by:  rnxpss2  5080  rnxpid  5081  ssxpbm  5082  ssxp2  5084  ssrnres  5089  funssxp  5404  fconst  5430  dff2  5681  fliftf  5821  tfrcllembfn  6382  frecuzrdgtcl  10443  cnconst2  14190  lmss  14203  exmidsbthrlem  15229
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