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Theorem ecexr 7634
 Description: A nonempty 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
StepHypRef Expression
1 n0i 3879 . . 3 (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2 snprc 4197 . . . . 5 𝐵 ∈ V ↔ {𝐵} = ∅)
3 imaeq2 5381 . . . . 5 ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
42, 3sylbi 206 . . . 4 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5 ima0 5400 . . . 4 (𝑅 “ ∅) = ∅
64, 5syl6eq 2660 . . 3 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
71, 6nsyl2 141 . 2 (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8 df-ec 7631 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
97, 8eleq2s 2706 1 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125   “ cima 5041  [cec 7627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631 This theorem is referenced by:  relelec  7674  ecdmn0  7676  erdisj  7681
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