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Theorem ecexr 8290
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 4282 . . 3 (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2 snprc 4638 . . . . 5 𝐵 ∈ V ↔ {𝐵} = ∅)
3 imaeq2 5912 . . . . 5 ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
42, 3sylbi 220 . . . 4 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5 ima0 5932 . . . 4 (𝑅 “ ∅) = ∅
64, 5syl6eq 2875 . . 3 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
71, 6nsyl2 143 . 2 (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8 df-ec 8287 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
97, 8eleq2s 2934 1 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  c0 4276  {csn 4550  cima 5545  [cec 8283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ec 8287
This theorem is referenced by:  relelec  8330  ecdmn0  8332  erdisj  8337  eqvreldisj  35958
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