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Theorem rnxpss 6183
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 5693 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 6168 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5911 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 6182 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 4014 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 4014 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3947   × cxp 5680  ccnv 5681  dom cdm 5682  ran crn 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693
This theorem is referenced by:  ssxpb  6185  ssrnres  6189  resssxp  6281  funssxp  6757  fconst  6788  dff2  7113  dff3  7114  fliftf  7327  frxp2  8158  frxp3  8165  marypha1lem  9476  marypha1  9477  dfac12lem2  10187  brdom4  10573  nqerf  10973  xptrrel  14985  lern  18616  cnconst2  23278  lmss  23293  tsmsxplem1  24148  causs  25317  i1f0  25707  itg10  25708  taylf  26388  noextendseq  27697  perpln2  28638  gsumpart  32923  locfinref  33656  sitg0  34180  heicant  37356  rntrclfvOAI  42348  rtrclex  43284  trclexi  43287  rtrclexi  43288  cnvtrcl0  43293  rntrcl  43295  brtrclfv2  43394  xphe  43448  rfovcnvf1od  43671
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