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Theorem rnxpss 5676
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 5229 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5661 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5432 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 5675 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3741 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3741 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3680   × cxp 5216  ccnv 5217  dom cdm 5218  ran crn 5219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229
This theorem is referenced by:  ssxpb  5678  ssrnres  5682  funssxp  6174  fconst  6204  dff2  6486  dff3  6487  fliftf  6680  marypha1lem  8455  marypha1  8456  dfac12lem2  9079  brdom4  9465  nqerf  9865  xptrrel  13841  lern  17347  cnconst2  21210  lmss  21225  tsmsxplem1  22078  causs  23217  i1f0  23574  itg10  23575  taylf  24235  perpln2  25726  locfinref  30138  sitg0  30638  noextendseq  32047  heicant  33676  rntrclfvOAI  37673  rtrclex  38343  trclexi  38346  rtrclexi  38347  cnvtrcl0  38352  rntrcl  38354  brtrclfv2  38438  rp-imass  38484  xphe  38494  rfovcnvf1od  38717
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