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Theorem rnxp 5229
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 4807 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5216 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4998 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2450 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 5015 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2474 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    =/= wne 2599   (/)c0 3704    X. cxp 4794   `'ccnv 4795   dom cdm 4796   ran crn 4797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804  df-dm 4806  df-rn 4807
This theorem is referenced by:  rnxpid  5232  ssxpb  5233  xpima  5241  unixp  5331  fconst5  6081  xpexr  6691  xpexr2  6692  fparlem3  6853  fparlem4  6854  frxp  6861  fodomr  7676  dfac5lem3  8507  fpwwe2lem13  9018  vdwlem8  14881  ramz  14926  gsumxp  17551  xkoccn  20576  txindislem  20590  cnextf  21023  metustexhalf  21513  ovolctb  22385  axlowdimlem13  24926  axlowdim1  24931  imadifxp  28158  sibf0  29119  ovoliunnfl  31889  voliunnfl  31891
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