Theorem xpid11 4920

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 4897	. . 3 |- ( ( A X. A ) = ( B X. B ) -> dom ( A X. A ) = dom ( B X. B ) )
2		dmxpid 4918	. . 3 |- dom ( A X. A ) = A
3		dmxpid 4918	. . 3 |- dom ( B X. B ) = B
4	1, 2, 3	3eqtr3g 2263	. 2 |- ( ( A X. A ) = ( B X. B ) -> A = B )
5		xpeq12 4712	. . 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 4691   dom cdm 4693

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-dm 4703

This theorem is referenced by:  intopsn  13314