Theorem rnxpid 4755

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 4754	. 2 |- ran ( A X. A ) C_ A
2		opelxp 4374	. . . . . 6 |- ( <. x , x >. e. ( A X. A ) <-> ( x e. A /\ x e. A ) )
3		anidm 376	. . . . . 6 |- ( ( x e. A /\ x e. A ) <-> x e. A )
4	2, 3	bitri 173	. . . . 5 |- ( <. x , x >. e. ( A X. A ) <-> x e. A )
5		opeq1 3549	. . . . . . . . 9 |- ( x = y -> <. x , x >. = <. y , x >. )
6	5	eleq1d 2106	. . . . . . . 8 |- ( x = y -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
7	6	equcoms 1594	. . . . . . 7 |- ( y = x -> ( <. x , x >. e. ( A X. A ) <-> <. y , x >. e. ( A X. A ) ) )
8	7	biimpd 132	. . . . . 6 |- ( y = x -> ( <. x , x >. e. ( A X. A ) -> <. y , x >. e. ( A X. A ) ) )
9	8	spimev 1741	. . . . 5 |- ( <. x , x >. e. ( A X. A ) -> E. y <. y , x >. e. ( A X. A ) )
10	4, 9	sylbir 125	. . . 4 |- ( x e. A -> E. y <. y , x >. e. ( A X. A ) )
11		vex 2560	. . . . 5 |- x e. _V
12	11	elrn2 4576	. . . 4 |- ( x e. ran ( A X. A ) <-> E. y <. y , x >. e. ( A X. A ) )
13	10, 12	sylibr 137	. . 3 |- ( x e. A -> x e. ran ( A X. A ) )
14	13	ssriv 2949	. 2 |- A C_ ran ( A X. A )
15	1, 14	eqssi 2961	1 |- ran ( A X. A ) = A

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   <.cop 3378   X. cxp 4343   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by: (None)