Theorem ecexr 6772

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 5105	. . . . 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 6769	. . . 4 |- [ B ] R = ( R " { B } )
4	2, 3	eleq2s 2327	. . 3 |- ( A e. [ B ] R -> E. x e. { B } x R A )
5		df-rex 2526	. . . 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 3706	. . . . . 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 2820	. 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 2203   E.wrex 2521   _Vcvv 2813   {csn 3689   class class class wbr 4109   "cima 4752   [cec 6765

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-ec 6769

This theorem is referenced by:  relelec  6809  ecdmn0m  6811