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Theorem rnxpss 5724
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 5277 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5709 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5480 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 5723 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3776 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3776 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3715   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  ssxpb  5726  ssrnres  5730  funssxp  6222  fconst  6252  dff2  6534  dff3  6535  fliftf  6728  marypha1lem  8504  marypha1  8505  dfac12lem2  9158  brdom4  9544  nqerf  9944  xptrrel  13920  lern  17426  cnconst2  21289  lmss  21304  tsmsxplem1  22157  causs  23296  i1f0  23653  itg10  23654  taylf  24314  perpln2  25805  locfinref  30217  sitg0  30717  noextendseq  32126  heicant  33757  rntrclfvOAI  37756  rtrclex  38426  trclexi  38429  rtrclexi  38430  cnvtrcl0  38435  rntrcl  38437  brtrclfv2  38521  rp-imass  38567  xphe  38577  rfovcnvf1od  38800
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