Theorem ecexr 6785

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 5110	. . . . 5 |- ( A e. ( R " { B } ) -> ( A e. ( R " { B } ) <-> E. x e. { B } x R A ) )
2	1	ibi 176	. . . 4 |- ( A e. ( R " { B } ) -> E. x e. { B } x R A )
3		df-ec 6782	. . . 4 |- [ B ] R = ( R " { B } )
4	2, 3	eleq2s 2329	. . 3 |- ( A e. [ B ] R -> E. x e. { B } x R A )
5		df-rex 2528	. . . 4 |- ( E. x e. { B } x R A <-> E. x ( x e. { B } /\ x R A ) )
6		simpl 109	. . . . . 6 |- ( ( x e. { B } /\ x R A ) -> x e. { B } )
7		velsn 3711	. . . . . 6 |- ( x e. { B } <-> x = B )
8	6, 7	sylib 122	. . . . 5 |- ( ( x e. { B } /\ x R A ) -> x = B )
9	8	eximi 1649	. . . 4 |- ( E. x ( x e. { B } /\ x R A ) -> E. x x = B )
10	5, 9	sylbi 121	. . 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 2822	. 2 |- ( B e. _V <-> E. x x = B )
13	11, 12	sylibr 134	1 |- ( A e. [ B ] R -> B e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   _Vcvv 2815   {csn 3694   class class class wbr 4114   "cima 4757   [cec 6778

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-ec 6782

This theorem is referenced by:  relelec  6822  ecdmn0m  6824