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Theorem rnxp 5483
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 5049 . . 3 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5470 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5247 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
41, 3eqtri 2632 . 2 ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5 dmxp 5265 . 2 (𝐴 ≠ ∅ → dom (𝐵 × 𝐴) = 𝐵)
64, 5syl5eq 2656 1 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wne 2780  c0 3874   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049
This theorem is referenced by:  rnxpid  5486  ssxpb  5487  xpima  5495  unixp  5585  fconst5  6376  xpexr  6999  xpexr2  7000  fparlem3  7166  fparlem4  7167  frxp  7174  fodomr  7996  dfac5lem3  8831  fpwwe2lem13  9343  vdwlem8  15530  ramz  15567  gsumxp  18198  xkoccn  21232  txindislem  21246  cnextf  21680  metustexhalf  22171  ovolctb  23065  axlowdimlem13  25634  axlowdim1  25639  imadifxp  28796  sibf0  29723  ovoliunnfl  32621  voliunnfl  32623
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