Theorem rnxpid 5055

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 5052	. 2 |- ran ( A X. A ) C_ A
2		opelxp 4650	. . . . . 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 3774	. . . . . . . . 9 |- ( x = y -> <. x , x >. = <. y , x >. )
6	5	eleq1d 2244	. . . . . . . 8 |- ( x = y -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
7	6	equcoms 1706	. . . . . . 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 1859	. . . . 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 2738	. . . . 5 |- x e. _V
12	11	elrn2 4862	. . . 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 3157	. 2 |- A C_ ran ( A X. A )
15	1, 14	eqssi 3169	1 |- ran ( A X. A ) = A

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1490   e. wcel 2146   <.cop 3592   X. cxp 4618   ran crn 4621

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631

This theorem is referenced by: (None)