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Theorem rnxpss 6017
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 5554 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 6002 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5761 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 6016 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3987 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3987 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3919   × cxp 5541  ccnv 5542  dom cdm 5543  ran crn 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-xp 5549  df-rel 5550  df-cnv 5551  df-dm 5553  df-rn 5554
This theorem is referenced by:  ssxpb  6019  ssrnres  6023  funssxp  6524  fconst  6554  dff2  6854  dff3  6855  fliftf  7058  marypha1lem  8890  marypha1  8891  dfac12lem2  9564  brdom4  9946  nqerf  10346  xptrrel  14338  lern  17833  cnconst2  21886  lmss  21901  tsmsxplem1  22756  causs  23900  i1f0  24289  itg10  24290  taylf  24954  perpln2  26503  locfinref  31135  sitg0  31631  noextendseq  33201  heicant  35004  rntrclfvOAI  39488  rtrclex  40173  trclexi  40176  rtrclexi  40177  cnvtrcl0  40182  rntrcl  40184  brtrclfv2  40284  rp-imass  40329  xphe  40339  rfovcnvf1od  40562
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