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Theorem dmxpid 4768
Description: The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.)
Assertion
Ref Expression
dmxpid dom (𝐴 × 𝐴) = 𝐴

Proof of Theorem dmxpid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4553 . . 3 (𝐴 × 𝐴) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)}
21dmeqi 4748 . 2 dom (𝐴 × 𝐴) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)}
3 elex2 2705 . . . 4 (𝑦𝐴 → ∃𝑥 𝑥𝐴)
43rgen 2488 . . 3 𝑦𝐴𝑥 𝑥𝐴
5 dmopab3 4760 . . 3 (∀𝑦𝐴𝑥 𝑥𝐴 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)} = 𝐴)
64, 5mpbi 144 . 2 dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)} = 𝐴
72, 6eqtri 2161 1 dom (𝐴 × 𝐴) = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1332  wex 1469  wcel 1481  wral 2417  {copab 3996   × cxp 4545  dom cdm 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-dm 4557
This theorem is referenced by:  dmxpin  4769  xpid11  4770  sqxpeq0  4970  xpider  6508  psmetdmdm  12532  xmetdmdm  12564
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