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Theorem rnxpss 5485
Description: The range of a Cartesian 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 5049 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5470 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5247 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 5484 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3598 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3598 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3540   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049
This theorem is referenced by:  ssxpb  5487  ssrnres  5491  funssxp  5974  fconst  6004  dff2  6279  dff3  6280  fliftf  6465  marypha1lem  8222  marypha1  8223  dfac12lem2  8849  brdom4  9233  nqerf  9631  xptrrel  13567  lern  17048  cnconst2  20897  lmss  20912  tsmsxplem1  21766  causs  22904  i1f0  23260  itg10  23261  taylf  23919  perpln2  25406  locfinref  29236  sitg0  29735  heicant  32614  rntrclfvOAI  36272  rtrclex  36943  trclexi  36946  rtrclexi  36947  cnvtrcl0  36952  rntrcl  36954  brtrclfv2  37038  rp-imass  37085  xphe  37095  rfovcnvf1od  37318
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