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Theorem rnxpss 6193
Description: The range of a Cartesian product is included in its 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 5699 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 6178 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5917 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 6192 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 4029 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 4029 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3962   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699
This theorem is referenced by:  ssxpb  6195  ssrnres  6199  resssxp  6291  funssxp  6764  fconst  6794  dff2  7118  dff3  7119  fliftf  7334  frxp2  8167  frxp3  8174  marypha1lem  9470  marypha1  9471  dfac12lem2  10182  brdom4  10567  nqerf  10967  xptrrel  15015  lern  18648  cnconst2  23306  lmss  23321  tsmsxplem1  24176  causs  25345  i1f0  25735  itg10  25736  taylf  26416  noextendseq  27726  perpln2  28733  gsumpart  33042  locfinref  33801  sitg0  34327  heicant  37641  rntrclfvOAI  42678  rtrclex  43606  trclexi  43609  rtrclexi  43610  cnvtrcl0  43615  rntrcl  43617  brtrclfv2  43716  xphe  43770  rfovcnvf1od  43993
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