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Theorem ecexr 7732
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 3912 . . 3 (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2 snprc 4244 . . . . 5 𝐵 ∈ V ↔ {𝐵} = ∅)
3 imaeq2 5450 . . . . 5 ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
42, 3sylbi 207 . . . 4 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5 ima0 5469 . . . 4 (𝑅 “ ∅) = ∅
64, 5syl6eq 2670 . . 3 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
71, 6nsyl2 142 . 2 (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8 df-ec 7729 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
97, 8eleq2s 2717 1 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  {csn 4168  cima 5107  [cec 7725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ec 7729
This theorem is referenced by:  relelec  7772  ecdmn0  7774  erdisj  7779
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