Theorem dmxpid 4980

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 4757	. . 3 ⊢ (𝐴 × 𝐴) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)}
2	1	dmeqi 4959	. 2 ⊢ dom (𝐴 × 𝐴) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)}
3		elex2 2832	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
4	3	rgen 2597	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴
5		dmopab3 4971	. . 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 2255	1 ⊢ dom (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  {copab 4172   × cxp 4749  dom cdm 4751

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-dm 4761

This theorem is referenced by:  dmxpin  4981  xpid11  4982  sqxpeq0  5188  xpider  6842  psmetdmdm  15238  xmetdmdm  15270