Theorem rnxpid 4805

Description: The range of a square cross product. (Contributed by FL, 17-May-2010.)

Assertion

Ref	Expression
rnxpid	|- ran ( A X. A ) = A

Proof of Theorem rnxpid

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnxpss 4804	. 2 |- ran ( A X. A ) C_ A
2		opelxp 4420	. . . . . 6 |- ( <. x , x >. e. ( A X. A ) <-> ( x e. A /\ x e. A ) )
3		anidm 388	. . . . . 6 |- ( ( x e. A /\ x e. A ) <-> x e. A )
4	2, 3	bitri 182	. . . . 5 |- ( <. x , x >. e. ( A X. A ) <-> x e. A )
5		opeq1 3590	. . . . . . . . 9 |- ( x = y -> <. x , x >. = <. y , x >. )
6	5	eleq1d 2151	. . . . . . . 8 |- ( x = y -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
7	6	equcoms 1636	. . . . . . 7 |- ( y = x -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
8	7	biimpd 142	. . . . . 6 |- ( y = x -> ( <. x , x >. e. ( A X. A ) -> <. y , x >. e. ( A X. A ) ) )
9	8	spimev 1784	. . . . 5 |- ( <. x , x >. e. ( A X. A ) -> E. y <. y , x >. e. ( A X. A ) )
10	4, 9	sylbir 133	. . . 4 |- ( x e. A -> E. y <. y , x >. e. ( A X. A ) )
11		vex 2613	. . . . 5 |- x e. _V
12	11	elrn2 4624	. . . 4 |- ( x e. ran ( A X. A ) <-> E. y <. y , x >. e. ( A X. A ) )
13	10, 12	sylibr 132	. . 3 |- ( x e. A -> x e. ran ( A X. A ) )
14	13	ssriv 3012	. 2 |- A C_ ran ( A X. A )
15	1, 14	eqssi 3024	1 |- ran ( A X. A ) = A

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434   <.cop 3419   X. cxp 4389   ran crn 4392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402

This theorem is referenced by: (None)