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Theorem dmxpin 5501
 Description: The domain of the intersection of two square Cartesian products. Unlike dmin 5487, equality holds. (Contributed by NM, 29-Jan-2008.)
Assertion
Ref Expression
dmxpin dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)

Proof of Theorem dmxpin
StepHypRef Expression
1 inxp 5410 . . 3 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴𝐵) × (𝐴𝐵))
21dmeqi 5480 . 2 dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴𝐵) × (𝐴𝐵))
3 dmxpid 5500 . 2 dom ((𝐴𝐵) × (𝐴𝐵)) = (𝐴𝐵)
42, 3eqtri 2782 1 dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∩ cin 3714   × cxp 5264  dom cdm 5266 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-dm 5276 This theorem is referenced by: (None)
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