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Theorem xpid11 5946
Description: The Cartesian square is a one-to-one construction. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
xpid11 ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)

Proof of Theorem xpid11
StepHypRef Expression
1 dmeq 5917 . . 3 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
2 dmxpid 5944 . . 3 dom (𝐴 × 𝐴) = 𝐴
3 dmxpid 5944 . . 3 dom (𝐵 × 𝐵) = 𝐵
41, 2, 33eqtr3g 2798 . 2 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵)
5 xpeq12 5714 . . 3 ((𝐴 = 𝐵𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
65anidms 566 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
74, 6impbii 209 1 ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537   × cxp 5687  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699
This theorem is referenced by:  intopsn  18680  grporn  30550  ismndo2  37861  rngosn3  37911  rngomndo  37922
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