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Theorem rnxp 5437
Description: The range of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
Assertion
Ref Expression
rnxp  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )

Proof of Theorem rnxp
StepHypRef Expression
1 df-rn 5010 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5424 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5204 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2496 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 5221 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2520 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    =/= wne 2662   (/)c0 3785    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  rnxpid  5440  ssxpb  5441  xpima  5449  unixp  5540  fconst5  6118  xpexr  6724  xpexr2  6725  fparlem3  6885  fparlem4  6886  frxp  6893  fodomr  7668  dfac5lem3  8506  fpwwe2lem13  9020  vdwlem8  14365  ramz  14402  gsumxp  16807  gsumxpOLD  16809  xkoccn  19883  txindislem  19897  cnextf  20329  metustexhalfOLD  20829  metustexhalf  20830  ovolctb  21664  axlowdimlem13  23961  axlowdim1  23966  imadifxp  27159  sibf0  27944  ovoliunnfl  29661  voliunnfl  29663
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