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Theorem dmxpid 4844
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 4629 . . 3 (𝐴 × 𝐴) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)}
21dmeqi 4824 . 2 dom (𝐴 × 𝐴) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)}
3 elex2 2753 . . . 4 (𝑦𝐴 → ∃𝑥 𝑥𝐴)
43rgen 2530 . . 3 𝑦𝐴𝑥 𝑥𝐴
5 dmopab3 4836 . . 3 (∀𝑦𝐴𝑥 𝑥𝐴 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)} = 𝐴)
64, 5mpbi 145 . 2 dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐴)} = 𝐴
72, 6eqtri 2198 1 dom (𝐴 × 𝐴) = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  wral 2455  {copab 4060   × cxp 4621  dom cdm 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-dm 4633
This theorem is referenced by:  dmxpin  4845  xpid11  4846  sqxpeq0  5048  xpider  6600  psmetdmdm  13484  xmetdmdm  13516
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