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Theorem rnxp 5722
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 5277 . . 3 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 5709 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 5480 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
41, 3eqtri 2782 . 2 ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5 dmxp 5499 . 2 (𝐴 ≠ ∅ → dom (𝐵 × 𝐴) = 𝐵)
64, 5syl5eq 2806 1 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wne 2932  c0 4058   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  rnxpid  5725  ssxpb  5726  xpima  5734  unixp  5829  fconst5  6635  xpexr  7271  xpexr2  7272  fparlem3  7447  fparlem4  7448  frxp  7455  fodomr  8276  dfac5lem3  9138  fpwwe2lem13  9656  vdwlem8  15894  ramz  15931  gsumxp  18575  xkoccn  21624  txindislem  21638  cnextf  22071  metustexhalf  22562  ovolctb  23458  axlowdimlem13  26033  axlowdim1  26038  imadifxp  29721  sibf0  30705  ovoliunnfl  33764  voliunnfl  33766
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