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Theorem rnxpid 6075
Description: The range of a Cartesian square. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
rnxpid ran (𝐴 × 𝐴) = 𝐴

Proof of Theorem rnxpid
StepHypRef Expression
1 rn0 5834 . . 3 ran ∅ = ∅
2 xpeq2 5611 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐴) = (𝐴 × ∅))
3 xp0 6060 . . . . 5 (𝐴 × ∅) = ∅
42, 3eqtrdi 2796 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
54rneqd 5846 . . 3 (𝐴 = ∅ → ran (𝐴 × 𝐴) = ran ∅)
6 id 22 . . 3 (𝐴 = ∅ → 𝐴 = ∅)
71, 5, 63eqtr4a 2806 . 2 (𝐴 = ∅ → ran (𝐴 × 𝐴) = 𝐴)
8 rnxp 6072 . 2 (𝐴 ≠ ∅ → ran (𝐴 × 𝐴) = 𝐴)
97, 8pm2.61ine 3030 1 ran (𝐴 × 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  c0 4262   × cxp 5588  ran crn 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-cnv 5598  df-dm 5600  df-rn 5601
This theorem is referenced by:  sofld  6089  fpwwe2lem12  10399  ustimasn  23378  utopbas  23385  restutop  23387  ovoliunlem1  24664  metideq  31839  poimirlem3  35776  mblfinlem1  35810  rtrclex  41195
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