Theorem dmxpid 4978

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 X. 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 4755	. . 3 |- ( A X. A ) = { <. y , x >. | ( y e. A /\ x e. A ) }
2	1	dmeqi 4957	. 2 |- dom ( A X. A ) = dom { <. y , x >. | ( y e. A /\ x e. A ) }
3		elex2 2830	. . . 4 |- ( y e. A -> E. x x e. A )
4	3	rgen 2595	. . 3 |- A. y e. A E. x x e. A
5		dmopab3 4969	. . 3 |- ( A. y e. A E. x x e. A <-> dom { <. y , x >. | ( y e. A /\ x e. A ) } = A )
6	4, 5	mpbi 145	. 2 |- dom { <. y , x >. | ( y e. A /\ x e. A ) } = A
7	2, 6	eqtri 2253	1 |- dom ( A X. A ) = A

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   {copab 4170   X. cxp 4747   dom cdm 4749

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-dm 4759

This theorem is referenced by:  dmxpin  4979  xpid11  4980  sqxpeq0  5186  xpider  6840  psmetdmdm  15189  xmetdmdm  15221