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Theorem ecexr 6047
Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecexr (A [B]𝑅B V)

Proof of Theorem ecexr
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elimag 4615 . . . . 5 (A (𝑅 “ {B}) → (A (𝑅 “ {B}) ↔ x {B}x𝑅A))
21ibi 165 . . . 4 (A (𝑅 “ {B}) → x {B}x𝑅A)
3 df-ec 6044 . . . 4 [B]𝑅 = (𝑅 “ {B})
42, 3eleq2s 2129 . . 3 (A [B]𝑅x {B}x𝑅A)
5 df-rex 2306 . . . 4 (x {B}x𝑅Ax(x {B} x𝑅A))
6 simpl 102 . . . . . 6 ((x {B} x𝑅A) → x {B})
7 elsn 3382 . . . . . 6 (x {B} ↔ x = B)
86, 7sylib 127 . . . . 5 ((x {B} x𝑅A) → x = B)
98eximi 1488 . . . 4 (x(x {B} x𝑅A) → x x = B)
105, 9sylbi 114 . . 3 (x {B}x𝑅Ax x = B)
114, 10syl 14 . 2 (A [B]𝑅x x = B)
12 isset 2555 . 2 (B V ↔ x x = B)
1311, 12sylibr 137 1 (A [B]𝑅B V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  wrex 2301  Vcvv 2551  {csn 3367   class class class wbr 3755  cima 4291  [cec 6040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044
This theorem is referenced by:  relelec  6082  ecdmn0m  6084
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