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Theorem rnxpss 5783
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 5323 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5768 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5528 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 5782 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 3831 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 3831 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3769   × cxp 5310  ccnv 5311  dom cdm 5312  ran crn 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  ssxpb  5785  ssrnres  5789  funssxp  6276  fconst  6306  dff2  6597  dff3  6598  fliftf  6793  marypha1lem  8581  marypha1  8582  dfac12lem2  9254  brdom4  9640  nqerf  10040  xptrrel  14062  lern  17540  cnconst2  21416  lmss  21431  tsmsxplem1  22284  causs  23424  i1f0  23795  itg10  23796  taylf  24456  perpln2  25962  locfinref  30424  sitg0  30924  noextendseq  32333  heicant  33933  rntrclfvOAI  38040  rtrclex  38707  trclexi  38710  rtrclexi  38711  cnvtrcl0  38716  rntrcl  38718  brtrclfv2  38802  rp-imass  38847  xphe  38857  rfovcnvf1od  39080
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