Theorem dmxpin 4952

Description: The domain of the intersection of two square Cartesian products. Unlike dmin 4937, equality holds. (Contributed by NM, 29-Jan-2008.)

Assertion

Ref	Expression
dmxpin	|- dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B )

Proof of Theorem dmxpin

Step	Hyp	Ref	Expression
1		inxp 4862	. . 3 |- ( ( A X. A ) i^i ( B X. B ) ) = ( ( A i^i B ) X. ( A i^i B ) )
2	1	dmeqi 4930	. 2 |- dom ( ( A X. A ) i^i ( B X. B ) ) = dom ( ( A i^i B ) X. ( A i^i B ) )
3		dmxpid 4951	. 2 |- dom ( ( A i^i B ) X. ( A i^i B ) ) = ( A i^i B )
4	2, 3	eqtri 2250	1 |- dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B )

Syntax hints:   = wceq 1395   i^i cin 3197   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-rex 2514  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-rel 4730  df-dm 4733

This theorem is referenced by: (None)