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Theorem rnxp 5273
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 4856 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5260 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 5046 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2463 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 5063 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2487 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    =/= wne 2611   (/)c0 3642    X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846
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 4418  ax-nul 4426  ax-pr 4536
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856
This theorem is referenced by:  rnxpid  5276  ssxpb  5277  xpima  5285  unixp  5375  fconst5  5940  xpexr  6523  xpexr2  6524  fparlem3  6679  fparlem4  6680  frxp  6687  fodomr  7467  dfac5lem3  8300  fpwwe2lem13  8814  vdwlem8  14054  ramz  14091  gsumxp  16473  gsumxpOLD  16475  xkoccn  19197  txindislem  19211  cnextf  19643  metustexhalfOLD  20143  metustexhalf  20144  ovolctb  20978  axlowdimlem13  23205  axlowdim1  23210  imadifxp  25944  sibf0  26725  ovoliunnfl  28438  voliunnfl  28440
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