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Theorem xpid11 5772
 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 5742 . . 3 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
2 dmxpid 5770 . . 3 dom (𝐴 × 𝐴) = 𝐴
3 dmxpid 5770 . . 3 dom (𝐵 × 𝐵) = 𝐵
41, 2, 33eqtr3g 2856 . 2 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵)
5 xpeq12 5548 . . 3 ((𝐴 = 𝐵𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
65anidms 570 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
74, 6impbii 212 1 ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   × cxp 5521  dom cdm 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-dm 5533 This theorem is referenced by:  intopsn  17876  grporn  28348  ismndo2  35463  rngosn3  35513  rngomndo  35524
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