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Theorem rnxp 6024
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 × 𝐵) = 𝐵)

Proof of Theorem rnxp
StepHypRef Expression
1 df-rn 5564 . . 3 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 6011 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5771 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
41, 3eqtri 2848 . 2 ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5 dmxp 5797 . 2 (𝐴 ≠ ∅ → dom (𝐵 × 𝐴) = 𝐵)
64, 5syl5eq 2872 1 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wne 3020  c0 4294   × cxp 5551  ccnv 5552  dom cdm 5553  ran crn 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564
This theorem is referenced by:  rnxpid  6027  ssxpb  6028  xpima  6036  unixp  6130  fconst5  6967  xpexr  7614  xpexr2  7615  fparlem3  7803  fparlem4  7804  frxp  7814  fodomr  8660  djuexb  9330  dfac5lem3  9543  fpwwe2lem13  10056  vdwlem8  16316  ramz  16353  gsumxp  19018  xkoccn  22145  txindislem  22159  cnextf  22592  metustexhalf  23083  ovolctb  24008  axlowdimlem13  26656  axlowdim1  26661  imadifxp  30268  sibf0  31480  ovoliunnfl  34803  voliunnfl  34805
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