Theorem ecexr 6427

Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
ecexr	|- ( A e. [ B ] R -> B e. _V )

Proof of Theorem ecexr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimag 4880	. . . . 5 |- ( A e. ( R " { B } ) -> ( A e. ( R " { B } ) <-> E. x e. { B } x R A ) )
2	1	ibi 175	. . . 4 |- ( A e. ( R " { B } ) -> E. x e. { B } x R A )
3		df-ec 6424	. . . 4 |- [ B ] R = ( R " { B } )
4	2, 3	eleq2s 2232	. . 3 |- ( A e. [ B ] R -> E. x e. { B } x R A )
5		df-rex 2420	. . . 4 |- ( E. x e. { B } x R A <-> E. x ( x e. { B } /\ x R A ) )
6		simpl 108	. . . . . 6 |- ( ( x e. { B } /\ x R A ) -> x e. { B } )
7		velsn 3539	. . . . . 6 |- ( x e. { B } <-> x = B )
8	6, 7	sylib 121	. . . . 5 |- ( ( x e. { B } /\ x R A ) -> x = B )
9	8	eximi 1579	. . . 4 |- ( E. x ( x e. { B } /\ x R A ) -> E. x x = B )
10	5, 9	sylbi 120	. . 3 |- ( E. x e. { B } x R A -> E. x x = B )
11	4, 10	syl 14	. 2 |- ( A e. [ B ] R -> E. x x = B )
12		isset 2687	. 2 |- ( B e. _V <-> E. x x = B )
13	11, 12	sylibr 133	1 |- ( A e. [ B ] R -> B e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   E.wrex 2415   _Vcvv 2681   {csn 3522   class class class wbr 3924   "cima 4537   [cec 6420

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-ec 6424

This theorem is referenced by:  relelec  6462  ecdmn0m  6464