Theorem xpid11 5926

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

Step	Hyp	Ref	Expression
1		dmeq 5897	. . 3 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
2		dmxpid 5924	. . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴
3		dmxpid 5924	. . 3 ⊢ dom (𝐵 × 𝐵) = 𝐵
4	1, 2, 3	3eqtr3g 2821	. 2 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵)
5		xpeq12 5683	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
6	5	anidms 577	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
7	4, 6	impbii 212	1 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   × cxp 5656  dom cdm 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-dm 5668

This theorem is used by:  intopsn  18882  grporn  31180  ismndo2  39074  rngosn3  39124  rngomndo  39135