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Theorem rnxpid 5808
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 5610 . . 3 ran ∅ = ∅
2 xpeq2 5363 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3 xp0 5793 . . . . 5 (𝐴 × ∅) = ∅
42, 3syl6eq 2877 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
54rneqd 5585 . . 3 (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6 id 22 . . 3 (𝐴 = ∅ → 𝐴 = ∅)
71, 5, 63eqtr4a 2887 . 2 (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8 rnxp 5805 . 2 (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
97, 8pm2.61ine 3082 1 ran (𝐴 × 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  c0 4144   × cxp 5340  ran crn 5343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-dm 5352  df-rn 5353
This theorem is referenced by:  sofld  5822  fpwwe2lem13  9779  ustimasn  22402  utopbas  22409  restutop  22411  ovoliunlem1  23668  metideq  30470  poimirlem3  33949  mblfinlem1  33983  rtrclex  38758
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