Theorem dmxpid 4959

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 4737	. . 3 |- ( A X. A ) = { <. y , x >. | ( y e. A /\ x e. A ) }
2	1	dmeqi 4938	. 2 |- dom ( A X. A ) = dom { <. y , x >. | ( y e. A /\ x e. A ) }
3		elex2 2820	. . . 4 |- ( y e. A -> E. x x e. A )
4	3	rgen 2586	. . 3 |- A. y e. A E. x x e. A
5		dmopab3 4950	. . 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 2252	1 |- dom ( A X. A ) = A

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2202   A.wral 2511   {copab 4154   X. cxp 4729   dom cdm 4731

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-dm 4741

This theorem is referenced by:  dmxpin  4960  xpid11  4961  sqxpeq0  5167  xpider  6818  psmetdmdm  15118  xmetdmdm  15150