Theorem xpid11 4901

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 4878	. . 3 |- ( ( A X. A ) = ( B X. B ) -> dom ( A X. A ) = dom ( B X. B ) )
2		dmxpid 4899	. . 3 |- dom ( A X. A ) = A
3		dmxpid 4899	. . 3 |- dom ( B X. B ) = B
4	1, 2, 3	3eqtr3g 2261	. 2 |- ( ( A X. A ) = ( B X. B ) -> A = B )
5		xpeq12 4694	. . 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 1373   X. cxp 4673   dom cdm 4675

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-dm 4685

This theorem is referenced by:  intopsn  13199