Theorem rnxpid 4981

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 4978	. 2 |- ran ( A X. A ) C_ A
2		opelxp 4577	. . . . . 6 |- ( <. x , x >. e. ( A X. A ) <-> ( x e. A /\ x e. A ) )
3		anidm 394	. . . . . 6 |- ( ( x e. A /\ x e. A ) <-> x e. A )
4	2, 3	bitri 183	. . . . 5 |- ( <. x , x >. e. ( A X. A ) <-> x e. A )
5		opeq1 3713	. . . . . . . . 9 |- ( x = y -> <. x , x >. = <. y , x >. )
6	5	eleq1d 2209	. . . . . . . 8 |- ( x = y -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
7	6	equcoms 1685	. . . . . . 7 |- ( y = x -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
8	7	biimpd 143	. . . . . 6 |- ( y = x -> ( <. x , x >. e. ( A X. A ) -> <. y , x >. e. ( A X. A ) ) )
9	8	spimev 1834	. . . . 5 |- ( <. x , x >. e. ( A X. A ) -> E. y <. y , x >. e. ( A X. A ) )
10	4, 9	sylbir 134	. . . 4 |- ( x e. A -> E. y <. y , x >. e. ( A X. A ) )
11		vex 2692	. . . . 5 |- x e. _V
12	11	elrn2 4789	. . . 4 |- ( x e. ran ( A X. A ) <-> E. y <. y , x >. e. ( A X. A ) )
13	10, 12	sylibr 133	. . 3 |- ( x e. A -> x e. ran ( A X. A ) )
14	13	ssriv 3106	. 2 |- A C_ ran ( A X. A )
15	1, 14	eqssi 3118	1 |- ran ( A X. A ) = A

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   <.cop 3535   X. cxp 4545   ran crn 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by: (None)