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Theorem rnxpss 5282
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  X.  B )  C_  B

Proof of Theorem rnxpss
StepHypRef Expression
1 df-rn 4863 . 2  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5267 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5053 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
4 dmxpss 5281 . . 3  |-  dom  ( B  X.  A )  C_  B
53, 4eqsstri 3398 . 2  |-  dom  `' ( A  X.  B
)  C_  B
61, 5eqsstri 3398 1  |-  ran  ( A  X.  B )  C_  B
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3340    X. cxp 4850   `'ccnv 4851   dom cdm 4852   ran crn 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-xp 4858  df-rel 4859  df-cnv 4860  df-dm 4862  df-rn 4863
This theorem is referenced by:  ssxpb  5284  ssrnres  5288  funssxp  5583  fconst  5608  dff2  5867  dff3  5868  fliftf  6020  marypha1lem  7695  marypha1  7696  dfac12lem2  8325  brdom4  8709  nqerf  9111  lern  15407  cnconst2  18899  lmss  18914  tsmsxplem1  19739  causs  20821  i1f0  21177  itg10  21178  taylf  21838  sitg0  26744  heicant  28438
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