Theorem ecexr 6802

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 5125	. . . . 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 6799	. . . 4 |- [ B ] R = ( R " { B } )
4	2, 3	eleq2s 2333	. . 3 |- ( A e. [ B ] R -> E. x e. { B } x R A )
5		df-rex 2534	. . . 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 3722	. . . . . 6 |- ( x e. { B } <-> x = B )
8	6, 7	sylib 122	. . . . 5 |- ( ( x e. { B } /\ x R A ) -> x = B )
9	8	eximi 1653	. . . 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 2828	. 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 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   {csn 3705   class class class wbr 4125   "cima 4772   [cec 6795

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-ec 6799

This theorem is referenced by:  relelec  6839  ecdmn0m  6841