Theorem dmxpid 4825

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.

Step	Hyp	Ref	Expression
1		df-xp 4610	. . 3 ⊢ (𝐴 × 𝐴) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)}
2	1	dmeqi 4805	. 2 ⊢ dom (𝐴 × 𝐴) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)}
3		elex2 2742	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
4	3	rgen 2519	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴
5		dmopab3 4817	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴)
6	4, 5	mpbi 144	. 2 ⊢ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴
7	2, 6	eqtri 2186	1 ⊢ dom (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  {copab 4042   × cxp 4602  dom cdm 4604

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-dm 4614

This theorem is referenced by:  dmxpin  4826  xpid11  4827  sqxpeq0  5027  xpider  6572  psmetdmdm  12964  xmetdmdm  12996