Theorem ecexr 6442

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 4893	. . . . 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 6439	. . . 4 |- [ B ] R = ( R " { B } )
4	2, 3	eleq2s 2235	. . 3 |- ( A e. [ B ] R -> E. x e. { B } x R A )
5		df-rex 2423	. . . 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 3549	. . . . . 6 |- ( x e. { B } <-> x = B )
8	6, 7	sylib 121	. . . . 5 |- ( ( x e. { B } /\ x R A ) -> x = B )
9	8	eximi 1580	. . . 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 2695	. 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 1332   E.wex 1469   e. wcel 1481   E.wrex 2418   _Vcvv 2689   {csn 3532   class class class wbr 3937   "cima 4550   [cec 6435

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-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-ec 6439

This theorem is referenced by:  relelec  6477  ecdmn0m  6479