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Theorem rnxpid 5807
Description: The range of a square Cartesian product. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
rnxpid ran (𝐴 × 𝐴) = 𝐴

Proof of Theorem rnxpid
StepHypRef Expression
1 rn0 5609 . . 3 ran ∅ = ∅
2 xpeq2 5362 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3 xp0 5792 . . . . 5 (𝐴 × ∅) = ∅
42, 3syl6eq 2876 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
54rneqd 5584 . . 3 (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6 id 22 . . 3 (𝐴 = ∅ → 𝐴 = ∅)
71, 5, 63eqtr4a 2886 . 2 (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8 rnxp 5804 . 2 (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
97, 8pm2.61ine 3081 1 ran (𝐴 × 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1658  c0 4143   × cxp 5339  ran crn 5342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-xp 5347  df-rel 5348  df-cnv 5349  df-dm 5351  df-rn 5352
This theorem is referenced by:  sofld  5821  fpwwe2lem13  9778  ustimasn  22401  utopbas  22408  restutop  22410  ovoliunlem1  23667  metideq  30480  poimirlem3  33955  mblfinlem1  33989  rtrclex  38764
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