Theorem xpid11 4900

Description: The Cartesian product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
xpid11	|- ( ( A X. A ) = ( B X. B ) <-> A = B )

Proof of Theorem xpid11

Step	Hyp	Ref	Expression
1		dmeq 4877	. . 3 |- ( ( A X. A ) = ( B X. B ) -> dom ( A X. A ) = dom ( B X. B ) )
2		dmxpid 4898	. . 3 |- dom ( A X. A ) = A
3		dmxpid 4898	. . 3 |- dom ( B X. B ) = B
4	1, 2, 3	3eqtr3g 2260	. 2 |- ( ( A X. A ) = ( B X. B ) -> A = B )
5		xpeq12 4693	. . 3 |- ( ( A = B /\ A = B ) -> ( A X. A ) = ( B X. B ) )
6	5	anidms 397	. 2 |- ( A = B -> ( A X. A ) = ( B X. B ) )
7	4, 6	impbii 126	1 |- ( ( A X. A ) = ( B X. B ) <-> A = B )

Syntax hints:   <-> wb 105   = wceq 1372   X. cxp 4672   dom cdm 4674

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-dm 4684

This theorem is referenced by:  intopsn  13170