Theorem dmxpid 4951

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 4729	. . 3 |- ( A X. A ) = { <. y , x >. | ( y e. A /\ x e. A ) }
2	1	dmeqi 4930	. 2 |- dom ( A X. A ) = dom { <. y , x >. | ( y e. A /\ x e. A ) }
3		elex2 2817	. . . 4 |- ( y e. A -> E. x x e. A )
4	3	rgen 2583	. . 3 |- A. y e. A E. x x e. A
5		dmopab3 4942	. . 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 2250	1 |- dom ( A X. A ) = A

Syntax hints:   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   {copab 4147   X. cxp 4721   dom cdm 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-dm 4733

This theorem is referenced by:  dmxpin  4952  xpid11  4953  sqxpeq0  5158  xpider  6770  psmetdmdm  15038  xmetdmdm  15070