Theorem dmxpid 4998

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 4775	. . 3 |- ( A X. A ) = { <. y , x >. | ( y e. A /\ x e. A ) }
2	1	dmeqi 4977	. 2 |- dom ( A X. A ) = dom { <. y , x >. | ( y e. A /\ x e. A ) }
3		elex2 2838	. . . 4 |- ( y e. A -> E. x x e. A )
4	3	rgen 2603	. . 3 |- A. y e. A E. x x e. A
5		dmopab3 4989	. . 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 2259	1 |- dom ( A X. A ) = A

Syntax hints:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   {copab 4186   X. cxp 4767   dom cdm 4769

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-dm 4779

This theorem is referenced by:  dmxpin  4999  xpid11  5000  sqxpeq0  5206  xpider  6870  psmetdmdm  15348  xmetdmdm  15380