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Theorem rnxpss 5446
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 5019 . 2  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5431 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5214 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
4 dmxpss 5445 . . 3  |-  dom  ( B  X.  A )  C_  B
53, 4eqsstri 3529 . 2  |-  dom  `' ( A  X.  B
)  C_  B
61, 5eqsstri 3529 1  |-  ran  ( A  X.  B )  C_  B
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3471    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019
This theorem is referenced by:  ssxpb  5448  ssrnres  5452  funssxp  5750  fconst  5777  dff2  6044  dff3  6045  fliftf  6214  marypha1lem  7911  marypha1  7912  dfac12lem2  8541  brdom4  8925  nqerf  9325  lern  15981  cnconst2  19910  lmss  19925  tsmsxplem1  20780  causs  21862  i1f0  22219  itg10  22220  taylf  22881  perpln2  24213  locfinref  27997  sitg0  28463  heicant  30211  xptrrel  37876  rp-imass  37896
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