Theorem rnxpid 5162

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 5159	. 2 |- ran ( A X. A ) C_ A
2		opelxp 4748	. . . . . 6 |- ( <. x , x >. e. ( A X. A ) <-> ( x e. A /\ x e. A ) )
3		anidm 396	. . . . . 6 |- ( ( x e. A /\ x e. A ) <-> x e. A )
4	2, 3	bitri 184	. . . . 5 |- ( <. x , x >. e. ( A X. A ) <-> x e. A )
5		opeq1 3856	. . . . . . . . 9 |- ( x = y -> <. x , x >. = <. y , x >. )
6	5	eleq1d 2298	. . . . . . . 8 |- ( x = y -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
7	6	equcoms 1754	. . . . . . 7 |- ( y = x -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
8	7	biimpd 144	. . . . . 6 |- ( y = x -> ( <. x , x >. e. ( A X. A ) -> <. y , x >. e. ( A X. A ) ) )
9	8	spimev 1907	. . . . 5 |- ( <. x , x >. e. ( A X. A ) -> E. y <. y , x >. e. ( A X. A ) )
10	4, 9	sylbir 135	. . . 4 |- ( x e. A -> E. y <. y , x >. e. ( A X. A ) )
11		vex 2802	. . . . 5 |- x e. _V
12	11	elrn2 4965	. . . 4 |- ( x e. ran ( A X. A ) <-> E. y <. y , x >. e. ( A X. A ) )
13	10, 12	sylibr 134	. . 3 |- ( x e. A -> x e. ran ( A X. A ) )
14	13	ssriv 3228	. 2 |- A C_ ran ( A X. A )
15	1, 14	eqssi 3240	1 |- ran ( A X. A ) = A

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   <.cop 3669   X. cxp 4716   ran crn 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by: (None)