Theorem xpid11 4961

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 4937	. . 3 |- ( ( A X. A ) = ( B X. B ) -> dom ( A X. A ) = dom ( B X. B ) )
2		dmxpid 4959	. . 3 |- dom ( A X. A ) = A
3		dmxpid 4959	. . 3 |- dom ( B X. B ) = B
4	1, 2, 3	3eqtr3g 2287	. 2 |- ( ( A X. A ) = ( B X. B ) -> A = B )
5		xpeq12 4750	. . 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 1398   X. cxp 4729   dom cdm 4731

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-dm 4741

This theorem is referenced by:  intopsn  13511