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Theorem rnxpss 4938
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 4518 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 4925 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 4708 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 4937 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3097 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3097 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff set class
Syntax hints:  wss 3039   × cxp 4505  ccnv 4506  dom cdm 4507  ran crn 4508
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  rnxpss2  4940  rnxpid  4941  ssxpbm  4942  ssxp2  4944  ssrnres  4949  funssxp  5260  fconst  5286  dff2  5530  fliftf  5666  tfrcllembfn  6220  frecuzrdgtcl  10136  cnconst2  12308  lmss  12321  exmidsbthrlem  13051
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