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Theorem rnxp 5781
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 5323 . . 3 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5768 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5528 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
41, 3eqtri 2821 . 2 ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5 dmxp 5547 . 2 (𝐴 ≠ ∅ → dom (𝐵 × 𝐴) = 𝐵)
64, 5syl5eq 2845 1 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wne 2971  c0 4115   × cxp 5310  ccnv 5311  dom cdm 5312  ran crn 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  rnxpid  5784  ssxpb  5785  xpima  5793  unixp  5887  fconst5  6700  xpexr  7341  xpexr2  7342  fparlem3  7516  fparlem4  7517  frxp  7524  fodomr  8353  dfac5lem3  9234  fpwwe2lem13  9752  vdwlem8  16025  ramz  16062  gsumxp  18690  xkoccn  21751  txindislem  21765  cnextf  22198  metustexhalf  22689  ovolctb  23598  axlowdimlem13  26191  axlowdim1  26196  imadifxp  29931  sibf0  30912  ovoliunnfl  33940  voliunnfl  33942
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