Theorem dmxpid 4953

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 4731	. . 3 |- ( A X. A ) = { <. y , x >. | ( y e. A /\ x e. A ) }
2	1	dmeqi 4932	. 2 |- dom ( A X. A ) = dom { <. y , x >. | ( y e. A /\ x e. A ) }
3		elex2 2819	. . . 4 |- ( y e. A -> E. x x e. A )
4	3	rgen 2585	. . 3 |- A. y e. A E. x x e. A
5		dmopab3 4944	. . 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 1397   E.wex 1540   e. wcel 2202   A.wral 2510   {copab 4149   X. cxp 4723   dom cdm 4725

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-dm 4735

This theorem is referenced by:  dmxpin  4954  xpid11  4955  sqxpeq0  5160  xpider  6774  psmetdmdm  15047  xmetdmdm  15079