Theorem dmxpid 4955

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 4733	. . 3 ⊢ (𝐴 × 𝐴) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)}
2	1	dmeqi 4934	. 2 ⊢ dom (𝐴 × 𝐴) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)}
3		elex2 2818	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
4	3	rgen 2584	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴
5		dmopab3 4946	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴)
6	4, 5	mpbi 145	. 2 ⊢ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴
7	2, 6	eqtri 2251	1 ⊢ dom (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2201  ∀wral 2509  {copab 4150   × cxp 4725  dom cdm 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-xp 4733  df-dm 4737

This theorem is referenced by:  dmxpin  4956  xpid11  4957  sqxpeq0  5162  xpider  6780  psmetdmdm  15077  xmetdmdm  15109